Adv Alegbra HW Solutions 65

Adv Alegbra HW Solutions 65 - 65(iii Prove that ker = Z(G...

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65 (iii) Prove that ker γ = Z ( G ) . Solution. Absent. (iv) Prove that im γ C Aut ( G ) . Solution. Absent. 2.93 If G is a group, prove that Aut ( G ) ={ 1 } if and only if | G |≤ 2. Solution. Instructors are cautioned: assume that G is f nite. The exercise is, in fact, true as stated, but it wants Zorn s lemma at one stage (see the following argument). If there is a G with a / Z ( G ) , then conjugation by a is a nontrivial automorphism; therefore, G is abelian. The map x 7→− x is an automor- phism of G ; if it is trivial, then x =− x for all x G . Thus, we may assume that that G is an abelian group (which we now write additively) with 2 x = 0; that is, G is a vector space over F 2 (the f eld with 2 ele- ments). If G is f nite-dimensional and dim ( G ) 2, then any nonsingular matrix other than the identity corresponds to a nontrivial automorphism. For example, if v 1 ,v 2 ,...,v n is a basis, then there is an automorphism which interchanges v 1 and v 2 and which f xes v 3 ,...,v n .I f G is in f nite,
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