Math 111A HW3 Homework 3
Computational Questions
You don’t need to prove the answers to these questions, but you should show your work to the
extent feasible.
1. Deduce the multiplication table for Q8 from the following information:
iv) ) 1 commutes with all elements of Q8.
2. Find the image and kernel of each group homomorphism. (You may assume the maps are
homomorphisms.)
(a) f : Z → Z /n Z given by f(x) = [x].
(b) g : D6 → S3 given by g(r) = (123), g(s) = (12).
(c) h : D20 → D10 given by h(r) = r, h(s) = sr.
Proof Questions
Your answers to these questions should be proofs. Problems labeled “DF” are from the text
book, Dummit and Foote.
3. Let G be a group. Let H ? G be a subset. Let H1, H2 ≤ G be subgroups.
(a) Prove that H is a subgroup if and only if the following two conditions hold:
1) e ∈ H
2) ?x, y ∈ H, xy??
1
∈ H.
(b) Prove that H1 ∩ H2 is a subgroup of G.
(c) Prove that H1 ∪ H2 is a subgroup of G if and only if H1 ? H2 or H2 ? H1.
4. Let G be a group, and let S ? G be a subset.
(a) Let H be a subgroup of G. Prove that if S ? H then hSi ? H.
1(b) Prove that hSi = \ H.
S?H≤G
Remark: This problem shows that hSi is the smallest subgroup of G which contains S.
5. Let G, H be a groups, and let f : G → H be a group homomorphism.
(a) Prove that e?
1
= e.
(b) Prove that f(eG) = eH.
(c) Let g ∈ G. Prove that f(g??
1
) = f(g)??
1
(d) If f is bijective, prove that f??
1
: H → G is also a group homomorphism.
6. Let G, H be groups, let x ∈ G, and let f : G → H be a group homomorphism.
(a) Suppose |x| = n < ∞. If x
k
= e for some k = 0, prove that n | k.
(b) If |x| = n < ∞, prove that |f(x)| | n.
(c) If f is an isomorphism, prove |f(x)| = |x|.
7. Prove that any homomorphism from D6 to Z /3 Z is the trivial homomorphism.
8. Let G, H be groups and let f : G → H be a group homomorphism. Let G
0
≤ G and H
0
≤ H
be subgroups.
(a) Prove that f(G
0
) is a subgroup of H.
(b) Prove that f??
1
(H
0
) is a subgroup of G.
9. Find all subgroups of S3. Prove that your list is complete.
10. Let G, H be fifinite groups, and suppose (|G|, |H|) = 1. Prove that the only homomorphism
from G to H is the trivial homomorphism.
11. DF §1.6, Exercise 4
12. Consider the following groups of order 8:
a) Z /8 Z
b) Z /4 Z × Z /2 Z
c) Z /2 Z × Z /2 Z × Z /2 Z
d) D8
e) Q8
Prove that none of these groups are isomorphic to each other. (We will prove later that every
group of order 8 is isomorphic to one of these fifive.)