Group Theory: Sheet 1

\begin{matrix}c \\ ◯ \end{matrix}G. C. Smith 2001

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1. Consider the group G = GL(n, ℂ) of n by n invertible matrices with complex entries, the group operation being matrix multiplication. A subset S of GL(n, ℂ) may or may not form a group in its own right (using the same group operation, and the same identity element I_{n}). Which of the following subsets of G form groups in their own right?

(a) The elements of G which are symmetric matrices.

(b) The elements X ∈ G such that XX̄^{T} = I_{n}. Here X̄ is obtained from X by replacing each entry of X by its complex conjugate. We use superscript T to denote taking the transpose of a matrix.

(c) The elements X in G which have finite order. (An element X has finite order if there is a positive integer m such that X^{m} is the identity element.)

2. Let G be the symmetric group Sym(ℤ), so G consist of all bijections from ℤ to ℤ. The order n of an element x in a group is the smallest natural number (if there are any) such that x^{n} = 1. Note that the identity element has order 1, and it is possible that there are no natural numbers with the property – hence the notion of infinite order mentioned above.

(a) Suppose that p, q are positive integers. Prove that there are elements θ, ψ ∈ G with θ of order p, ψ of order q, where θ commutes with ψ.

(b) Suppose that p, q are positive integers. Prove that there are elements θ, ψ ∈ G with θ of order p, ψ of order q, but θ does not commute with ψ.

(c) Give an example of an element ζ ∈ G of infinite order.

(d) Give examples of elements α, β ∈, each of infinite order, such that αβ has order 2.

(e) Suppose that r is a positive integer bigger than 2. Do there exist μ, ν ∈ G both of infinite order such that μν has order r?

(f) Do there exist σ, τ ∈ G both of infinite order such that σ, τ has infinite order?

(g) Suppose that η, ν ∈ G both have finite order. Does it follow that ην has finite order?

3. Let G be a group, and suppose that x, y ∈ G. Prove that xy and yx have the same order. This means that both have infinite order, or both have the same finite order.

4. Let G be a group, and suppose that x, y, z ∈ G. Prove that xyz, yzx and zxy must all have the same order. Compare this with the order of xzy.