Let H and K be subgroups of a group G so that for all h in H and
k in K there is a k' in K with hk = k'h. Proposition 2.3.2 shows
that HK is a group. Show that K is a normal subgroup of HK.
Find all the subgroups of the group of symmetries of a cube.
Show all steps.
Hint: Label the diagonals as 1, 2, 3, and 4 then consider the
rotations to get the subgroups.
Find two distinct subgroups of order 2 of the group D3 of
symmetries of an equilateral triangle. Explain why this fact alone
shows that D3 is not a cynic group.
Consider the group Z/24Z.
(a) Find the subgroups 〈21〉 and 〈10〉.
(b) Find all generators for the subgroup 〈21〉 ∩ 〈10〉.
(c) In general, what is a generator for 〈a〉 ∩ 〈b〉 in Z/nZ? Prove
your assertion.
(1) Let G be a group and H, K be subgroups of G.
(a) Show that if H is a normal subgroup, then HK = {xy|x ? H, y ?
K} is a
subgroup of G.
(b) Show that if H and K are both normal subgroups, then HK is also
a normal
subgroup.
(c) Give an example of subgroups H and K such that HK is not a
subgroup of G.