Unformatted text preview: G of order 2 is odd. In particular, G must contain an element of order 2. 5. For x an element of G , where G is a group, show that x and x-1 have the same order. 6. Find elements A , B and C in D 4 (the group of symmetries of the square) such that AB = BC but A 6 = C . 7. Assume H is a nonempty subset of ( G,? ) which is closed under the binary operation on G and is closed under inverses, i.e., for all h and k ∈ H , hk and h-1 ∈ H . Prove that H is a group under the operation ? restricted to H . ( H is called a subgroup of G ) 8. Let G be a group, and let y ∈ G have order m . If m = dt for some d ≥ 1, prove that y t has order d ....
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- Spring '11