Mathematics 1214: Introduction to Group Theory Tutorial exercise sheet 9 and G

Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 9
1. Let G1 and G2 be two groups.
Prove that if H1 ≤ G1 and H2 ≤ G2 , then H1 × H2 ≤ G1 × G2 .
[Here, H1 × H2 is the set {(h1 , h2 ) : h1 ∈ H1 , h2 ∈ H2 }.]
2. (a) Explain how the relationship Z ⊆ R allows us to identify Z × Z with a subset
of the plane P = R2 = R × R, and use this to “draw a picture” of Z × Z.
(b) Let G = Z × Z and let H = h(1, 2)i. Draw a picture of H, and draw a picture
of the right coset Ha where a = (−2, 3).
(c) The right cosets of H partition G. “Explain” this by drawing a picture. How
many (different) right cosets of H are there?
3. If G and H are groups with |G| > 1 and |H| > 1, prove that there are at least four
different subgroups of G × H.
[Hint: consider subgroups of the form S × T where S ≤ G and T ≤ H].
4. (a) Let G be a group. Prove that ∆ = {(g, g) : g ∈ G} is a subgroup of G × G.
(b) Find a counterexample to the following statement:
If G1 and G2 are groups, then
H ≤ G1 × G2 =⇒ H = H1 × H2 for some subgroups H1 ≤ G1 and H2 ≤ G2 .
[Hint: take G1 = G2 = Z2 and apply (a).]
5. Prove that if G is a finite group of prime order, then G is abelian.
6. A corollary to Lagrange’s theorem (Corollary 39) tells us that if n is a prime
number, then every group of order n is cyclic.
For each non-prime integer n with 4 ≤ n ≤ 14, give an example of a cyclic group
of order n, and a non-cyclic group of order n.
[Hint: the cyclic group should be easy. For the non-cyclic group, if n is even with
n ≥ 6 then you should be able to think of a non-abelian group of order n, and
then consider the remaining cases, n = 4 and n = 9 separately.]
[Remark: the case n = 15 is different. Even though 15 is not prime, it turns out
that every group of order 15 is cyclic!].