Some Comments on Exam 1
(1) Let ϕ : G → Ĝ denote a group homomorphism. Let K := {g ∈ G : ϕ(g) = ê}, where ê denotes the
identity element in Ĝ.
(a) Prove that K is a subgroup of G.
(b) Prove that K G.
We talked about this problem in class. You can read more in chapter 10.
(2) Let G := Z2 ⊕ Z9 .
(a) How many generators does G have?
(b) What are the generators of G?
(c) How many subgroups does G have?
(d) List all subgroups of G.
First note that G ' Z18 . You know how to answer all of the questions for a cyclic group. Next
use the (explicit) isomorphism Z18 ' Z2 ⊕ Z9 to determine the generators and subgroups in G.
(3) Let H denote the group
−1
0
0
1
1
,
0
0
−1
0
,
1
−1
0
0
,
−1
1
0
(a) Find all of the elements in H.
(b) Find all subgroups of H, and draw the complete lattice of subgroups.
(c) Is H abelian?
We did part of this problem in class. You should get 8 elements in the group, and you may note
that H is isomorphic to D4 . You should have 5 subgroups of order 2 and 3 subgroups of order 4.
Not all of the order 4 subgroups are cyclic. You can check your work by looking at the lattice of
subgroups for D4 .
(4) Prove that the permutations (12) and (12345) generate S5 .
This is an outline for Leo’s proof, organized a bit differently.
• We proved that any element of Sn can be expressed as a product of transpositions. So, one
approach is to prove that we can generate all 2-cycles in S5 from the two permutations given.
• Write τ := (12) and σ := (12345). We have:
σ −1 τ σ = (54321)(12)(12345)
= (51)
.
σ −2 τ σ 2 = (54321)2 (12)(12345)2
= (54321)(51)(12345)
= (45).
1
2
Continue conjugating τ by higher powers of σ, and you will obtain the transpositions:
(12), (23), (34), (45), (51). Note that you can generalize this argument to Sn for n > 5.
• We now try to generate the 2-cycle (14). We have (14) = (12)(23)(34)(23)(12). You should argue that the 2-cycle (1a), for any a, is given by: (1a) = (12)(23) · · · (a−1, a)(a−2, a−1) · · · (12).
• Finally, we want to generate an arbitrary 2-cycle (ab). This is just (1a)(1b)(1a).
• This shows that we can get all transpositions, and hence all permutations.
• You could, perhaps, condense the last 2 steps of this argument into one step. This theorem–and
this proof–generalizes to Sn . Could we take a different 2-cycle, along with (12345), and from
these generate all of S5 ?