Equianharmonic

In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.

The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small representations, and lists of all duplicates. (In removing duplicates it is useful to note that finite simple groups are determined by their orders, except that the group B_n(q) has the same order as C_n(q) for q odd, n > 2; and the groups A₈ = A₃(2) and A₂(4) both have orders 20160.)

Notation: n is a positive integer, q > 1 is a power of a prime number p, and is the order of some underlying finite field. The order of the outer automorphism group is written as d·f·g, where d is the order of the group of "diagonal automorphisms", f is the order of the (cyclic) group of "field automorphisms" (generated by a Frobenius automorphism), and g is the order of the group of "graph automorphisms" (coming from automorphisms of the Dynkin diagram).

Infinite families

Cyclic groups Z_p

Simplicity: Simple for p a prime number.

Order: p

Schur multiplier: Trivial.

Outer automorphism group: Cyclic of order p − 1.

Other names: Z/pZ

Remarks: These are the only simple groups that are not perfect.

A_n, n > 4, Alternating groups

Simplicity: Solvable for n < 5, otherwise simple.

Order: n!/2 when n > 1.

Schur multiplier: 2 for n = 5 or n > 7, 6 for n = 6 or 7; see Covering groups of the alternating and symmetric groups

Outer automorphism group: In general 2. Exceptions: for n = 1, n = 2, it is trivial, and for n = 6, it has order 4 (elementary abelian).

Other names: Alt_n.

There is an unfortunate conflict with the notation for the (unrelated) groups A_n(q), and some authors use various different fonts for A_n to distinguish them. In particular, in this article we make the distinction by setting the alternating groups A_n in Roman font and the Lie-type groups A_n(q) in italic.

Isomorphisms: A₁ and A₂ are trivial. A₃ is cyclic of order 3. A₄ is isomorphic to A₁(3) (solvable). A₅ is isomorphic to A₁(4) and to A₁(5). A₆ is isomorphic to A₁(9) and to the derived group B₂(2)'. A₈ is isomorphic to A₃(2).

Remarks: An index 2 subgroup of the symmetric group of permutations of n points when n > 1.

Chevalley groups A_n(q), B_n(q) n > 1, C_n(q) n > 2, D_n(q) n > 3

	Chevalley groups An(q) linear groups	Chevalley groups Bn(q) n > 1 orthogonal groups	Chevalley groups Cn(q) n > 2 symplectic groups	Chevalley groups Dn(q) n > 3 orthogonal groups
Simplicity:	A1(2) and A1(3) are solvable, the others are simple.	B2(2) is not simple but its derived group B2(2)′ is a simple subgroup of index 2; the others are simple.	All simple	All simple
Order:	${1 \over (n+1,q-1)}q^{n(n+1)/2}\prod _{i=1}^{n}(q^{i+1}-1)$	${1 \over (2,q-1)}q^{n^{2}}\prod _{i=1}^{n}(q^{2i}-1)$	${1 \over (2,q-1)}q^{n^{2}}\prod _{i=1}^{n}(q^{2i}-1)$	${1 \over (4,q^{n}-1)}q^{n(n-1)}(q^{n}-1)\prod _{i=1}^{n-1}(q^{2i}-1)$
Schur multiplier:	For the simple groups it is cyclic of order (n+1, q − 1) except for A1(4) (order 2), A1(9) (order 6), A2(2) (order 2), A2(4) (order 48, product of cyclic groups of orders 3, 4, 4), A3(2) (order 2).	(2,q − 1) except for B2(2) = S6 (order 2 for B2(2), order 6 for B2(2)′) and B3(2) (order 2) and B3(3) (order 6).	(2,q − 1) except for C3(2) (order 2).	The order is (4, qn − 1) (cyclic for n odd, elementary abelian for n even) except for D4(2) (order 4, elementary abelian).
Outer automorphism group:	(2, q − 1) ·f·1 for n = 1; (n+1, q − 1) ·f·2 for n > 1, where q = pf.	(2, q − 1) ·f·1 for q odd or n>2; (2, q − 1) ·f·2 if q is even and n=2, where q = pf.	(2, q − 1) ·f·1 where q = pf.	(2, q − 1) 2·f·S3 for n=4, (2, q − 1) 2·f·2 for n>4 even, (4, qn − 1)·f·2 for n odd, where q = pf, and S3 is the symmetric group of order 3! on 3 points.
Other names:	Projective special linear groups, PSLn+1(q), Ln+1(q), PSL(n+1,q)	O2n+1(q), Ω2n+1(q) (for q odd).	Projective symplectic group, PSp2n(q), PSpn(q) (not recommended), S2n(q), Abelian group (archaic).	O2n+(q), PΩ2n+(q). "Hypoabelian group" is an archaic name for this group in characteristic 2.
Isomorphisms:	A1(2) is isomorphic to the symmetric group on 3 points of order 6. A1(3) is isomorphic to the alternating group A4 (solvable). A1(4) and A1(5) are isomorphic, and are both isomorphic to the alternating group A5. A1(7) and A2(2) are isomorphic. A1(8) is isomorphic to the derived group 2G2(3)′. A1(9) is isomorphic to A6 and to the derived group B2(2)′. A3(2) is isomorphic to A8.	Bn(2m) is isomorphic to Cn(2m). B2(2) is isomorphic to the symmetric group on 6 points, and the derived group B2(2)′ is isomorphic to A1(9) and to A6. B2(3) is isomorphic to 2A3(22).	Cn(2m) is isomorphic to Bn(2m)
Remarks:	These groups are obtained from the general linear groups GLn+1(q) by taking the elements of determinant 1 (giving the special linear groups SLn+1(q)) and then quotienting out by the center.	This is the group obtained from the orthogonal group in dimension 2n+1 by taking the kernel of the determinant and spinor norm maps. B1(q) also exists, but is the same as A1(q). B2(q) has a non-trivial graph automorphism when q is a power of 2.	This group is obtained from the symplectic group in 2n dimensions by quotienting out the center. C1(q) also exists, but is the same as A1(q). C2(q) also exists, but is the same as B2(q).	This is the group obtained from the split orthogonal group in dimension 2n by taking the kernel of the determinant (or Dickson invariant in characteristic 2) and spinor norm maps and then killing the center. The groups of type D4 have an unusually large diagram automorphism group of order 6, containing the triality automorphism. D2(q) also exists, but is the same as A1(q)×A1(q). D3(q) also exists, but is the same as A3(q).

Chevalley groups E₆(q), E₇(q), E₈(q), F₄(q), G₂(q)

	Chevalley groups E6(q)	Chevalley groups E7(q)	Chevalley groups E8(q)	Chevalley groups F4(q)	Chevalley groups G2(q)
Simplicity:	All simple	All simple	All simple	All simple	G2(2) is not simple but its derived group G2(2)′ is a simple subgroup of index 2; the others are simple.
Order:	q36(q12 − 1)(q9 − 1)(q8 − 1)(q6 − 1)(q5 − 1)(q2 − 1)/(3,q − 1)	q63(q18 − 1)(q14 − 1)(q12 − 1)(q10 − 1)(q8 − 1)(q6 − 1)(q2 − 1)/(2,q − 1)	q120(q30−1)(q24−1)(q20−1)(q18−1)(q14−1)(q12−1)(q8−1)(q2−1)	q24(q12−1)(q8−1)(q6−1)(q2−1)	q6(q6−1)(q2−1)
Schur multiplier:	(3,q − 1)	(2,q − 1)	Trivial	Trivial except for F4(2) (order 2).	Trivial for the simple groups except for G2(3) (order 3) and G2(4) (order 2).
Outer automorphism group:	(3, q − 1) ·f·2 where q = pf.	(2, q − 1) ·f·1 where q = pf.	1·f·1 where q = pf.	1·f·1 for q odd, 1·f·2 for q even, where q = pf.	1·f·1 for q not a power of 3, 1·f·2 for q a power of 3, where q = pf.
Other names:	Exceptional Chevalley group	Exceptional Chevalley group	Exceptional Chevalley group	Exceptional Chevalley group	Exceptional Chevalley group
Isomorphisms:					The derived group G2(2)′ is isomorphic to 2A2(32).
Remarks:	Has two representations of dimension 27, and acts on the Lie algebra of dimension 78.	Has a representations of dimension 56, and acts on the corresponding Lie algebra of dimension 133.	It acts on the corresponding Lie algebra of dimension 248. E8(3) contains the Thompson simple group.	These groups act on 27 dimensional exceptional Jordan algebras, which gives them 26 dimensional representations. They also act on the corresponding Lie algebras of dimension 52. F4(q) has a non-trivial graph automorphism when q is a power of 2.	These groups are the automorphism groups of 8-dimensional Cayley algebras over finite fields, which gives them 7 dimensional representations. They also act on the corresponding Lie algebras of dimension 14. G2(q) has a non-trivial graph automorphism when q is a power of 3.

Steinberg groups ²A_n(q²) n > 1, ²D_n(q²) n > 3, ²E₆(q²), ³D₄(q³)

	Steinberg groups 2An(q2) n > 1 unitary groups	Steinberg groups 2Dn(q2) n > 3 orthogonal groups	Steinberg groups 2E6(q2)	Steinberg groups 3D4(q3)
Simplicity:	2A2(22) is solvable, the others are simple.	All simple	All simple	All simple
Order:	${1 \over (n+1,q+1)}q^{n(n+1)/2}\prod _{i=1}^{n}(q^{i+1}-(-1)^{i+1})$	${1 \over (4,q^{n}+1)}q^{n(n-1)}(q^{n}+1)\prod _{i=1}^{n-1}(q^{2i}-1)$	q36(q12−1)(q9+1)(q8−1)(q6−1)(q5+1)(q2−1)/(3,q+1)	q12(q8+q4+1)(q6−1)(q2−1)
Schur multiplier:	Cyclic of order (n + 1, q + 1) for the simple groups, except for 2A3(22) (order 2), 2A3(32) (order 36, product of cyclic groups of orders 3,3,4), 2A5(22) (order 12, product of cyclic groups of orders 2,2,3)	Cyclic of order (4, qn + 1)	(3, q + 1) except for 2E6(22) (order 12, product of cyclic groups of orders 2,2,3).	Trivial
Outer automorphism group:	(n+1, q + 1) ·f·1 where q2 = pf	(4, qn + 1) ·f·1 where q2 = pf	(3, q + 1) ·f·1 where q2 = pf.	1·f·1 where q3 = pf.
Other names:	Twisted Chevalley group, projective special unitary group, PSUn+1(q), PSU(n+1, q), Un+1(q), 2 An(q), 2An(q, q2)	2Dn(q), O2n−(q), PΩ2n−(q), twisted Chevalley group. "Hypoabelian group" is an archaic name for this group in characteristic 2.	2E6(q), twisted Chevalley group.	3D4(q), D42(q3), Twisted Chevalley groups.
Isomorphisms:	The solvable group 2A2(22) is isomorphic to an extension of the order 8 quaternion group by an elementary abelian group of order 9. 2A2(32) is isomorphic to the derived group G2(2)′. 2A3(22) is isomorphic to B2(3).
Remarks:	This is obtained from the unitary group in n+1 dimensions by taking the subgroup of elements of determinant 1 and then quotienting out by the center.	This is the group obtained from the non-split orthogonal group in dimension 2n by taking the kernel of the determinant (or Dickson invariant in characteristic 2) and spinor norm maps and then killing the center. 2D2(q2) also exists, but is the same as A1(q2). 2D3(q2) also exists, but is the same as 2A3(q2).	One of the exceptional double covers of 2E6(22) is a subgroup of the baby monster group, and the exceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group.	3D4(23) acts on the unique even 26 dimensional lattice of determinant 3 with no roots.

²B₂(2²ⁿ⁺¹) Suzuki groups

Simplicity: Simple for n ≥ 1. The group ²B₂(2) is solvable.

Order: q² (q² + 1) (q − 1) where q = 2²ⁿ⁺¹.

Schur multiplier: Trivial for n ≠ 1, elementary abelian of order 4 for ²B₂(8).

Outer automorphism group:

1·f·1

where f = 2n + 1.

Other names: Suz(2²ⁿ⁺¹), Sz(2²ⁿ⁺¹).

Isomorphisms: ²B₂(2) is the Frobenius group of order 20.

Remarks: Suzuki group are Zassenhaus groups acting on sets of size (2²ⁿ⁺¹)² + 1, and have 4 dimensional representations over the field with 2²ⁿ⁺¹ elements. They are the only non-cyclic simple groups whose order is not divisible by 3. They are not related to the sporadic Suzuki group.

²F₄(2²ⁿ⁺¹) Ree groups, Tits group

Simplicity: Simple for n ≥ 1. The derived group ²F₄(2)′ is simple of index 2 in ²F₄(2), and is called the Tits group, named for the Belgian mathematician Jacques Tits.

Order: q¹² (q⁶ + 1) (q⁴ − 1) (q³ + 1) (q − 1) where q = 2²ⁿ⁺¹.

The Tits group has order 17971200 = 2¹¹ · 3³ · 5² · 13.

Schur multiplier: Trivial for n ≥ 1 and for the Tits group.

Outer automorphism group:

1·f·1

where f = 2n + 1. Order 2 for the Tits group.

Remarks: Unlike the other simple groups of Lie type, the Tits group does not have a BN pair, though its automorphism group does so most authors count it as a sort of honorary group of Lie type.

²G₂(3²ⁿ⁺¹) Ree groups

Simplicity: Simple for n ≥ 1. The group ²G₂(3) is not simple, but its derived group ²G₂(3)′ is a simple subgroup of index 3.

Order: q³ (q³ + 1) (q − 1) where q = 3²ⁿ⁺¹

Schur multiplier: Trivial for n≥1 and for ²G₂(3)′.

Outer automorphism group:

1·f·1

where f = 2n + 1.

Other names: Ree(3²ⁿ⁺¹), R(3²ⁿ⁺¹), E₂^*(3²ⁿ⁺¹) .

Isomorphisms: The derived group ²G₂(3)′ is isomorphic to A₁(8).

Remarks: ²G₂(3²ⁿ⁺¹) has a doubly transitive permutation representation on 3^3(2n+1) + 1 points and acts on a 7-dimensional vector space over the field with 3²ⁿ⁺¹ elements.

Sporadic groups

Mathieu groups M₁₁, M₁₂, M₂₂, M₂₃, M₂₄

	Mathieu group M11	Mathieu group M12	Mathieu group M22	Mathieu group M23	Mathieu group M24
Order:	24 · 32 · 5 · 11=7920	26 · 33 · 5 · 11=95040	27 · 32 · 5 · 7 · 11 = 443520	27 · 32 · 5 · 7 · 11 · 23=10200960	210 · 33 · 5 · 7 · 11 · 23= 244823040
Schur multiplier:	Trivial	Order 2	Cyclic of order 12Template:Efn	Trivial	Trivial
Outer automorphism group:	Trivial	Order 2	Order 2	Trivial	Trivial
Remarks:	A 4-transitive permutation group on 11 points, and the point stabilizer in M12. The subgroup fixing a point is sometimes called M10, and has a subgroup of index 2 isomorphic to the alternating group A6.	A 5-transitive permutation group on 12 points.	A 3-transitive permutation group on 22 points.	A 4-transitive permutation group on 23 points, contained in M24.	A 5-transitive permutation group on 24 points.

Janko groups J₁, J₂, J₃, J₄

	Janko group J1	Janko group J2	Janko group J3	Janko group J4
Order:	23 · 3 · 5 · 7 · 11 · 19 = 175560	27 · 33 · 52 · 7 = 604800	27 · 35 · 5 · 17 · 19 = 50232960	221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 = 86775571046077562880
Schur multiplier:	Trivial	Order 2	Order 3	Trivial
Outer automorphism group:	Trivial	Order 2	Order 2	Trivial
Other names:	J(1), J(11)	Hall–Janko group, HJ
Remarks:	It is a subgroup of G2(11), and so has a 7 dimensional representation over the field with 11 elements.	It is the automorphism group of a rank 3 graph on 100 points, and is also contained in G2(4).	J3 seems unrelated to any other sporadic groups (or to anything else). Its triple cover has a 9 dimensional unitary representation over the field with 4 elements.	Has a 112 dimensional representation over the field with 2 elements.

Conway groups Co₁, Co₂, Co₃

	Conway group Co1	Conway group Co2	Conway group Co3
Order:	221 · 39 · 54 · 72 · 11 · 13 · 23 = 4157776806543360000	218 · 36 · 53 · 7 · 11 · 23 = 42305421312000	210 · 37 · 53 · 7 · 11 · 23 = 495766656000
Schur multiplier:	Order 2	Trivial	Trivial
Outer automorphism group:	Trivial	Trivial	Trivial
Other names:	·1	·2	·3, C3
Remarks:	The perfect double cover of Co1 is the automorphism group of the Leech lattice, and is sometimes denoted by ·0.	Subgroup of Co1; fixes a norm 4 vector in the Leech lattice.	Subgroup of Co1; fixes a norm 6 vector in the Leech lattice. It has a doubly transitive permutation representation on 276 points.

Fischer groups Fi₂₂, Fi₂₃, Fi₂₄'

	Fischer group Fi22	Fischer group Fi23	Fischer group Fi24'
Order:	217 · 39 · 52 · 7 · 11 · 13 = 64561751654400	218 · 313 · 52 · 7 · 11 · 13 · 17 · 23 = 4089470473293004800	221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 29 = 1255205709190661721292800
Schur multiplier:	Order 6	Trivial	Order 3
Outer automorphism group:	Order 2	Trivial	Order 2
Other names:	M(22)	M(23)	M(24)′, F3+
Remarks:	A 3-transposition group whose double cover is contained in Fi23.	A 3-transposition group contained in Fi24.	The triple cover is contained in the monster group.

Higman–Sims group HS

Order: 2⁹ · 3² · 5³· 7 · 11 = 44352000

Schur multiplier: Order 2.

Outer automorphism group: Order 2.

Remarks: It acts as a rank 3 permutation group on the Higman Sims graph with 100 points, and is contained in Co₃.

McLaughlin group McL

Order: 2⁷ · 3⁶ · 5³· 7 · 11 = 898128000

Schur multiplier: Order 3.

Outer automorphism group: Order 2.

Remarks: Acts as a rank 3 permutation group on the McLaughlin graph with 275 points, and is contained in Co₃.

Held group He

Order: 2¹⁰ · 3³ · 5²· 7³· 17 = 4030387200

Schur multiplier: Trivial.

Outer automorphism group: Order 2.

Other names: Held–Higman–McKay group, HHM, F₇, HTH

Remarks: Centralizes an element of order 7 in the monster group.

Rudvalis group Ru

Order: 2¹⁴ · 3³ · 5³· 7 · 13 · 29 = 145926144000

Schur multiplier: Order 2.

Outer automorphism group: Trivial.

Remarks: The double cover acts on a 28 dimensional lattice over the Gaussian integers.

Suzuki sporadic group Suz

Order: 2¹³ · 3⁷ · 5²· 7 · 11 · 13 = 448345497600

Schur multiplier: Order 6.

Outer automorphism group: Order 2.

Other names: Sz

Remarks: The 6 fold cover acts on a 12 dimensional lattice over the Eisenstein integers. It is not related to the Suzuki groups of Lie type.

O'Nan group O'N

Order: 2⁹ · 3⁴ · 5 · 7³ · 11 · 19 · 31 = 460815505920

Schur multiplier: Order 3.

Outer automorphism group: Order 2.

Other names: O'Nan–Sims group, O'NS, O–S

Remarks: The triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism.

Harada–Norton group HN

Order: 2¹⁴ · 3⁶ · 5⁶ · 7 · 11 · 19 = 273030912000000

Schur multiplier: Trivial.

Outer automorphism group: Order 2.

Other names: F₅, D

Remarks: Centralizes an element of order 5 in the monster group.

Lyons group Ly

Order: 2⁸ · 3⁷ · 5⁶ · 7 · 11 · 31 · 37 · 67 = 51765179004000000

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: Lyons–Sims group, LyS

Remarks: Has a 111 dimensional representation over the field with 5 elements.

Thompson group Th

Order: 2¹⁵ · 3¹⁰ · 5³ · 7² · 13 · 19 · 31 = 90745943887872000

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: F₃, E

Remarks: Centralizes an element of order 3 in the monster, and is contained in E₈(3), so has a 248-dimensional representation over the field with 3 elements.

Baby Monster group B

Order:

   2⁴¹ · 3¹³ · 5⁶ · 7² · 11 · 13 · 17 · 19 · 23 · 31 · 47

= 4154781481226426191177580544000000

Schur multiplier: Order 2.

Outer automorphism group: Trivial.

Other names: F₂

Remarks: The double cover is contained in the monster group. It has a representation of dimension 4371 over the complex numbers (with no nontrivial invariant product), and a representation of dimension 4370 over the field with 2 elements preserving a commutative but non-associative product.

Fischer–Griess Monster group M

Order:

   2⁴⁶ · 3²⁰ · 5⁹ · 7⁶ · 11² · 13³ · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71

= 808017424794512875886459904961710757005754368000000000

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: F₁, M₁, Monster group, Friendly giant, Fischer's monster.

Remarks: Contains all but 6 of the other sporadic groups as subquotients. Related to monstrous moonshine. The monster is the automorphism group of the 196884 dimensional Griess algebra and the infinite dimensional monster vertex operator algebra, and acts naturally on the monster Lie algebra.

Non-cyclic simple groups of small order

Order	Factored order	Group	Schur multiplier	Outer automorphism group
60	22 · 3 · 5	A5 = A1(4) = A1(5)	2	2
168	23 · 3 · 7	A1(7) = A2(2)	2	2
360	23 · 32 · 5	A6 = A1(9) = B2(2)′	6	2×2
504	23 · 32 · 7	A1(8) = 2G2(3)′	1	3
660	22 · 3 · 5 · 11	A1(11)	2	2
1092	22 · 3 · 7 · 13	A1(13)	2	2
2448	24 · 32 · 17	A1(17)	2	2
2520	23 · 32 · 5 · 7	A7	6	2
3420	22 · 32 · 5 · 19	A1(19)	2	2
4080	24 · 3 · 5 · 17	A1(16)	1	4
5616	24 · 33 · 13	A2(3)	1	2
6048	25 · 33 · 7	2A2(9) = G2(2)′	1	2
6072	23 · 3 · 11 · 23	A1(23)	2	2
7800	23 · 3 · 52 · 13	A1(25)	2	2×2
7920	24 · 32 · 5 · 11	M11	1	1
9828	22 · 33 · 7 · 13	A1(27)	2	6
12180	22 · 3 · 5 · 7 · 29	A1(29)	2	2
14880	25 · 3 · 5 · 31	A1(31)	2	2
20160	26 · 32 · 5 · 7	A3(2) = A8	2	2
20160	26 · 32 · 5 · 7	A2(4)	3×42	D12
25308	22 · 32 · 19 · 37	A1(37)	2	2
25920	26 · 34 · 5	2A3(4) = B2(3)	2	2
29120	26 · 5 · 7 · 13	2B2(8)	22	3
32736	25 · 3 · 11 · 31	A1(32)	1	5
34440	23 · 3 · 5 · 7 · 41	A1(41)	2	2
39732	22 · 3 · 7 · 11 · 43	A1(43)	2	2
51888	24 · 3 · 23 · 47	A1(47)	2	2
58800	24 · 3 · 52 · 72	A1(49)	2	22
62400	26 · 3 · 52 · 13	2A2(16)	1	4
74412	22 · 33 · 13 · 53	A1(53)	2	2
95040	26 · 33 · 5 · 11	M12	2	2

(Complete for orders less than 100,000)

Template:Harvtxt lists the 56 non-cyclic simple groups of order less than a million.

See also

• List of small groups

References

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Further reading

• Simple Groups of Lie Type by Roger W. Carter, ISBN 0-471-50683-4
• Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups." Oxford, England 1985.
• Daniel Gorenstein, Richard Lyons, Ronald Solomon The Classification of the Finite Simple Groups (volume 1), AMS, 1994 (volume 2), AMS,
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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Atlas of Finite Group Representations: contains representations and other data for many finite simple groups, including the sporadic groups.

External links

• Orders of non abelian simple groups up to order 10,000,000,000.