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 pgroup. 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
 KeithCornell
 Symmetric group, Solution., I8

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