8.2 Cauchy's Theorem
1. a. D4 = {1, a, a2, a3, b, ba, ba2, ba3} where o(a) = 4, o(b) = 2 and aba = b. The classes are {1}, {a, a3}, {a2}, {b, ba2}, {ba, ba3}. Since normal subgroups are of orders 1, 2, 4, 8, they are {1}, {1, a2}, {1, a, a2, a3}, {1, b, a2, ba2}, {1, a2, ba, ba3} and D4.
3. If am = g−1ag with g 
G, then
G, then
By induction we get 
for all k ≥ 0. Since G is finite, let gk = 1, k ≥ 1. Then 
whence 1 − mk = qn. This gives 1 = mk + qn, so gcd (m, n) = 1.
for all k ≥ 0. Since G is finite, let gk = 1, k ≥ 1. Then whence 1 − mk = qn. This gives 1 = mk + qn, so gcd (m, n) = 1.
5. Let H = class a1 ∪ . . . ∪ classan. Given g G and h H, let h classai. Then g−1hg classai ⊆ H, so g−1hg H. Then g−1Hg ⊆ H, as ...
G and h H, let h classai. Then g−1hg classai ⊆ H, so g−1hg H. Then g−1Hg ⊆ H, as ...Get Introduction to Abstract Algebra, Solutions Manual, 4th Edition now with the O’Reilly learning platform.
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