SECTION 4.2 Basics of Group Theory 239
a homomorphism of GL 2 (R) into the multiplicative group of non-
zero real numbers.- (Really the same as Exercise 3) Let G be any group and fix an
element x∈G. Prove that the mapping f : G → Gdefined by
settingf(g) =xgx−^1 is an isomorphism ofGonto itself. - LetAbe an Abeliangroup and letf :A→B be asurjective
homomorphism, where B is also a group. Prove that B is also
Abelian. - Letf :G→H be a homomorphism of groups and setK ={g∈
G|f(g) =eH}, whereeH is the identity ofH. Prove thatK is a
subgroup ofG.^13 - LetX={ 1 , 2 , 3 , ..., n}, wherenis a positive integer. Recall that
we have the group (2X,+), where, as usual, 2X is the power set of
Xand + is symmetric difference (see page 194). Definef : 2X→
{− 1 , 1 }(where{± 1 }is a group with respect to multiplication) by
setting
f(A) =
+1 if|A|is even
− 1 if|A|is odd.Prove thatf is a homomorphism.- LetGbe a group and definef :G→Gby settingf(g) =g−^1.
(a) Show thatf is a bijection.
(b) Under what circumstances isf a homomorphism?- Prove that the automorphism groups of the graphs on page 207
(each having four vertices) are not isomorphic. - Let R be the additive group of real numbers and assume that
f:R→Ris a function which satisfiesf(x−y) =f(x)−f(y), for
0 < x, y∈R. Prove thatf:R→Ris a homomorphism.
(^13) This subgroup ofGis usually called thekernelof the homomorphismf.