∗1. (10 points) Let A and B be subgroups of a group G. Recall from Assignment 1 that
A ∩ B is a subgroup of G, and hence of A.
(a) Show that
f : A/(A ∩ B) → G/B, a(A ∩ B) 7→ aB
is a well-defined map, i.e., is independent of any choice of coset representatives.
(b) Show that f is injective.
(c) Deduce that (A : A ∩ B) ≤ (G : B) in the case where (G : B) is finite.
(d) Suppose now that G = Sn for some integer n ≥ 2. Show that if A contains an odd
permutation, then exactly half the elements of A are odd
Hint: Use (c)

.
∗2. (7 points) Let G and H be groups.
(a) Show that if A is a subgroup of G, and B is a subgroup of H, then A × B is a
subgroup of G × H.
(b) Is is possible for G × H to have subgroups that are not of the kind described in
part (a)? Justify your answer with a suitable example or proof.
3. What are the possible orders of the elements of Z6 × Z9? How many elements achieve
the largest possible order?
4. (4 points) For a fixed positive integer m ≥ 2, consider the following subsets of D2m:
A = {id, r2
, r4
, · · · , r2m−2
, s, sr2
, sr4
, . . . , sr2m−2
}; B = {id, rm}.
(a) Show that A and B are subgroups of D2m.
(b) Show that A ∼= Dm and B ∼= Z2.
∗(c) Show that if m is odd, then D2m is the internal direct product of A and B (and
hence that D2m
∼= Dm × Z2).
∗5. (8 points) Show that if n ≥ 3 is odd, then Dn is not the internal direct product of
two non-trivial subgroups.
∗6. (7 points) Let H and N be subgroups of a group G.
(a) Show that if N is normal, then HN is a subgroup of G.
(b) Show that if both N and H are normal, then HN is also normal

DETAILED ASSIGNMENT

20200930063635hw21_1_