Group Theory : Homomorphism, Subgroups, Abelian Groups and Group Order
2.Let G be abelian of order n. If gcd(n;m) = 1, prove that f(g) = gm is an automorphism of G. (Note: Automorphism is just an isomorphism from G to itself.)
3. If f : Z7 ! Z5 is a homomorphism, prove that f(x) = 0 for all x 2 Z7.
4. Prove that in the group S10 every permutation of order 20 must be odd.
5. Suppose G is a group in which all subgroups are normal. Suppose further that x; y 2 G are such that gcd(jxj; jyj) = 1.
(a) Prove that x¡1y¡1xy 2 hxi hyi.
(b) Prove that xy = yx.
6. Let H be a subgroup of G, of ¯nite index [G : H] = k. Let C = fgH : g 2 Gg be the set of cosets of H in G.
(a) Prove that for x 2 G the function ¸x(gH) = xgH is a permutation of C.
(b) Let S(C) denote the group of permutations of the set C. Prove that the function F : G 7! S(C) given by F(x) = ¸x is a homomorphism.
(c) Prove that ker(F) · H.
(d) Prove that if jGj = n and n does not divide k!, then ker(F) 6= feg.
(e) Apply this to prove that if jGj = 99 and [G : H] = 9 then H must be a normal subgroup of G.
7. Let G be an abelian group of order 16. Write down all direct products of cyclic groups that could be isomorphic to G. If G has elements x and y, each of order 4, with x2 6= y2, can you determine G (up to isomorphism)?
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Math 3033 Group Theory
Holiday Assignment
◦◦◦
1. Let G be a group and x, y ∈ G. Prove that |x| = |x−1 |, |x−1 yx| = |y |
and |xy | = |yx|.
2. Let G be abelian of order n. If gcd(n, m) = 1, prove that f (g ) = g m is
an automorphism of G. (Note: Automorphism is just an isomorphism
from G to itself.)
3. If f : Z7 → Z5 is a homomorphism, prove that f (x) = 0 for all x ∈ Z7 .
4. Prove that in the group S10 every permutation of order 20 must be odd.
5. Suppose G is a group in which all subgroups are normal. Suppose
further that x, y ∈ G are such that gcd(|x|, |y |) = 1.
(a) Prove that x−1 y −1 xy ∈ x ∩ y .
(b) Prove that xy = yx.
6. Let H be a subgrou of G, of finite index [G : H ] = k . Let C = {gH :
g ∈ G} be the set of cosets of H in G.
(a) Prove that for x ∈ G the function λx (gH ) = xgH is a permutation
of C .
(b) Let S (C ) denote the group of permutations of the set C . Prove
that the function F : G → S (C ) given by F (x) = λx is a homo-
morphism.
(c) Prove that ker(F ) ≤ H .
(d) Prove that if |G| = n and n does not divide k !, then ker(F ) = {e}.
(e) Apply this to prove that if |G| = 99 and [G : H ] = 9 then H must
be a normal subgroup of G.
7. Let G be an abelian group of order 16. Write down all direct products
of cyclic groups that could be isomorphic to G. If G has elements x
and y , each of order 4, with x2 = y 2 , can you determine G (up to
isomorphism)?
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