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By Randall R. Holmes

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Therefore, y = xm = xqn+r = (xn )q xr = eq xr = xr ∈ S. (⊇) This follows immediately from the definition of x . ) Assume that xi = xj with 0 ≤ i ≤ j < n. Then 0 ≤ j − i < n and, arguing just as in the proof of (i), we get xj−i = e. Since ord(x) = n, it follows that j − i = 0, that is, j = i. This proves that the elements xi , 0 ≤ i < n, are distinct. 48 (iii) This follows from (i) and (ii). 6 Example Find the cyclic subgroup of Z12 generated by 3. 2 it was shown that ord(3) = 4. Put x = 3. According to part (ii) of the preceding theorem, we have 3 = x = {e, x, 2x, 3x} (additive notation) = {0, 3, 6, 9}.

For instance, {0, 2, 4, 6, 8, 10} = 2 . 1). 50 5 – Exercises 5–1 Let G be a group. The center of G, denoted Z(G), is the set of those elements of G that commute with every element of G: Z(G) = {z ∈ G | zx = xz for all x ∈ G}. Prove that Z(G) is a subgroup of G. 5–2 Fix n ∈ N. Let SLn (R) be the set of all n × n matrices over R having determinant 1: SLn (R) = {A ∈ Matn (R) | det(A) = 1}. Prove that SLn (R) is a subgroup of GLn (R) (= invertible n × n matrices over R). ) Hint: From linear algebra, we know that a square matrix is invertible if and only if its determinant is nonzero.

Use the fact that det(AB) = det(A) det(B) for A, B ∈ Matn (R). 5–3 (a) Find the order of the element 9 in Z15 . (b) Find the order of the matrix A = 0 −1 in the group Mat2×2 (R). 1 0 (c) Find the order of the matrix A = 0 −1 in the group GL2 (R). 1 0 5–4 (a) Compute 6 in the group Z15 . (b) Compute σ , where σ is the element of S4 given by σ= 1 2 3 4 . 3 1 4 2 51 5–5 Compute A in the group GL2 (R), where A = 1 1 . 0 1 5–6 Draw a subgroup diagram for the group Z18 including the subgroups {0}, 2 , 3 , 6 , 9 , and Z18 .

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Abstract Algebra I by Randall R. Holmes

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