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Unformatted text preview: last two because the f rst three are abelian while D 8 and Q are not abelian. Now I 8 = I 4 I 2 and I 8 = I 2 I 2 I 2 because I 8 has an element of order 8 and the other two groups do not. Similarly, I 4 I 2 = I 2 I 2 I 2 because I 4 I 2 has an element of order 4 and I 2 I 2 I 2 does not. Finally, D 8 = Q because Q has only 1 element of order 2 while D 8 has more than one element of order 2. 2.117 If p is a prime and G is a f nite group in which every element has order a power of p , prove that G is a p-group. Solution. If q is a prime divisor of | G | with q = p , then Cauchy s theorem gives an element in G of order q , contrary to the hypothesis....
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- Fall '11