|
|
| Line 1: |
Line 1: |
| In [[mathematics]], especially in the area of [[algebra]] known as [[group theory]], the '''holomorph''' of a [[group (mathematics)|group]] is a group which simultaneously contains (copies of) the group and its [[automorphism group]]. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform context. In group theory, for a group <math>G</math>, the holomorph of <math>G</math> denoted <math>\operatorname{Hol}(G)</math> can be described as a [[semidirect product]] or as a [[permutation group]].
| | Myrtle Benny is how I'm called and I really feel comfy when people use the complete name. The favorite hobby for my children and me is to play baseball but I haven't made a dime with it. Years in the past we moved to North Dakota. Bookkeeping is my occupation.<br><br>My web-site - [http://Www.Videokeren.com/blog/14593 std home test] |
| | |
| ==Hol(''G'') as a semi-direct product==
| |
| If <math>\operatorname{Aut}(G)</math> is the [[automorphism group]] of <math>G</math> then
| |
| :<math>\operatorname{Hol}(G)=G\rtimes \operatorname{Aut}(G)</math>
| |
| where the multiplication is given by
| |
| :<math>(g,\alpha)(h,\beta)=(g\alpha(h),\alpha\beta).</math> [Eq. 1]
| |
| | |
| Typically, a semidirect product is given in the form <math>G\rtimes_{\phi}A</math> where <math>G</math> and <math>A</math> are groups and <math>\phi:A\rightarrow \operatorname{Aut}(G)</math> is a [[homomorphism]] and where the multiplication of elements in the semi-direct product is given as
| |
| :<math>(g,a)(h,b)=(g\phi(a)(h),ab)</math>
| |
| which is [[well defined]], since <math>\phi(a)\in \operatorname{Aut}(G)</math> and therefore <math>\phi(a)(h)\in G</math>.
| |
| | |
| For the holomorph, <math>A=\operatorname{Aut}(G)</math> and <math>\phi</math> is the [[identity function|identity map]], as such we suppress writing <math>\phi</math> explicitly in the multiplication given in [Eq. 1] above.
| |
| | |
| For example,
| |
| * <math>G=C_3=\langle x\rangle=\{1,x,x^2\}</math> the [[cyclic group]] of order 3
| |
| * <math>\operatorname{Aut}(G)=\langle \sigma\rangle=\{1,\sigma\}</math> where <math>\sigma(x)=x^2</math>
| |
| * <math>\operatorname{Hol}(G)=\{(x^i,\sigma^j)\}</math> with the multiplication given by:
| |
| :<math>(x^{i_1},\sigma^{j_1})(x^{i_2},\sigma^{j_2}) = (x^{i_1+i_22^{^{j_1}}},\sigma^{j_1+j_2})</math> where the exponents of <math>x</math> are taken [[modular arithmetic|mod]] 3 and those of <math>\sigma</math> mod 2.
| |
| | |
| Observe, for example
| |
| :<math>(x,\sigma)(x^2,\sigma)=(x^{1+2\cdot2},\sigma^2)=(x^2,1)</math>
| |
| and note also that this group is not [[abelian group|abelian]], as <math>(x^2,\sigma)(x,\sigma)=(x,1)</math>, so that <math>\operatorname{Hol}(C_3)</math> is a [[non-abelian group]] of order 6 which, by basic group theory, must be [[isomorphic]] to the [[symmetric group]] <math>S_3</math>.
| |
| | |
| ==Hol(''G'') as a permutation group==
| |
| | |
| A group ''G'' acts naturally on itself by left and right multiplication, each giving rise to a [[group homomorphism|homomorphism]] from ''G'' into the [[symmetric group]] on the underlying set of ''G''. One homomorphism is defined as ''λ'': ''G'' → Sym(''G''), ''λ''(''g'')(''h'') = ''g''·''h''. That is, ''g'' is mapped to the [[permutation]] obtained by left multiplying each element of ''G'' by ''g''. Similarly, a second homomorphism ''ρ'': ''G'' → Sym(''G'') is defined by ''ρ''(''g'')(''h'') = ''h''·''g''<sup>−1</sup>, where the inverse ensures that ''ρ''(''g''·''h'')(''k'') = ''ρ''(''g'')(''ρ''(''h'')(''k'')). These homomorphisms are called the left and right [[regular representation]]s of ''G''. Each homomorphism is [[injective]], a fact referred to as [[Cayley's theorem]].
| |
| | |
| For example, if ''G'' = ''C''<sub>3</sub> = {1, ''x'', ''x''<sup>2</sup> } is a [[cyclic group]] of order three, then
| |
| * ''λ''(''x'')(1) = ''x''·1 = ''x'',
| |
| * ''λ''(''x'')(''x'') = ''x''·''x'' = ''x''<sup>2</sup>, and
| |
| * ''λ''(''x'')(''x''<sup>2</sup>) = ''x''·''x''<sup>2</sup> = 1,
| |
| so ''λ''(''x'') takes (1, ''x'', ''x''<sup>2</sup>) to (''x'', ''x''<sup>2</sup>, 1).
| |
| | |
| The image of ''λ'' is a subgroup of Sym(''G'') isomorphic to ''G'', and its [[normalizer]] in Sym(''G'') is defined to be the '''holomorph''' ''H'' of ''G''.
| |
| For each ''f'' in ''H'' and ''g'' in ''G'', there is an ''h'' in ''G'' such that ''f''·''λ''(''g'') = ''λ''(''h'')·''f''. If an element ''f'' of the holomorph fixes the [[identity element|identity]] of ''G'', then for ''1'' in ''G'', (''f''·''λ''(''g''))(''1'') = (''λ''(''h'')·''f'')(''1''), but the left hand side is ''f''(''g''), and the right side is ''h''. In other words, if ''f'' in ''H'' fixes the identity of ''G'', then for every ''g'' in ''G'', ''f''·''λ''(''g'') = ''λ''(''f''(''g''))·''f''. If ''g'', ''h'' are elements of ''G'', and ''f'' is an element of ''H'' fixing the identity of ''G'', then applying this equality twice to ''f''·''λ''(''g'')·''λ''(''h'') and once to the (equivalent) expression ''f''·''λ''(''g''·''h'') gives that ''f''(''g'')·''f''(''h'') = ''f''(''g''·''h''). That is, every element of ''H'' that fixes the identity of ''G'' is in fact an [[automorphism]] of ''G''. Such an ''f'' normalizes any ''λ''(''g''), and the only ''λ''(''g'') that fixes the identity is ''λ''(1). Setting ''A'' to be the [[stabilizer (group theory)]] of the identity, the subgroup generated by ''A'' and ''λ''(''G'') is [[semidirect product]] with [[normal subgroup]] ''λ''(''G'') and [[Complement (group theory)|complement]] ''A''. Since ''λ''(''G'') is [[transitive action|transitive]], the subgroup generated by ''λ''(''G'') and the point stabilizer ''A'' is all of ''H'', which shows the holomorph as a permutation group is isomorphic to the holomorph as semidirect product.
| |
| | |
| It is useful, but not directly relevant, that the [[centralizer]] of ''λ''(''G'') in Sym(''G'') is ''ρ''(''G''), their intersection is ''ρ''(Z(''G'')) = ''λ''(Z(''G'')), where Z(''G'') is the [[center (group theory)|center]] of ''G'', and that ''A'' is a common complement to both of these normal subgroups of ''H''.
| |
| | |
| ==Notes==
| |
| * ''ρ''(''G'') ∩ Aut(''G'') = 1
| |
| * Aut(''G'') normalizes ''ρ''(''G'') so that [[canonical form|canonically]] ''ρ''(''G'')Aut(''G'') ≅ ''G'' ⋊ Aut(''G'')
| |
| *<math>\operatorname{Inn}(G)\cong \operatorname{Im}(g\mapsto \lambda(g)\rho(g))</math> since ''λ''(''g'')''ρ''(''g'')(''h'') = ''ghg''<sup>−1</sup>
| |
| * ''K'' ≤ ''G'' is a [[characteristic subgroup]] if and only if ''λ''(''K'') ⊴ Hol(''G'')
| |
| | |
| ==References==
| |
| * {{Citation | last1=Hall | first1=Marshall, Jr. | author1-link=Marshall Hall (mathematician) | title=The theory of groups | publisher=Macmillan | id={{MathSciNet | id = 0103215}} | year=1959}}
| |
| | |
| [[Category:Group theory]]
| |
| [[Category:Group automorphisms]]
| |
Myrtle Benny is how I'm called and I really feel comfy when people use the complete name. The favorite hobby for my children and me is to play baseball but I haven't made a dime with it. Years in the past we moved to North Dakota. Bookkeeping is my occupation.
My web-site - std home test