Homework 09

# Homework 09 - G | &amp;amp;lt; . Prove that H must be...

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MATH 74 HOMEWORK 9 Due Wednesday April 24th at 3:10pm. Throughout, let G be a group. Many of the following problems are from (or adapted from) Herestein’s Topics in Algebra . (1) Let Z ( G ) be the center of G and suppose that G/Z ( G ) is cyclic. Prove that G is abelian. (2) Prove that any group of order 15 is cyclic. (3) If H < G , and H has exactly two distinct cosets in G , prove that H is normal. (4) If N is a normal subgroup of G and H is any subgroup, prove that NH is a subgroup of G . If H is normal, prove that NH is too. (5) Suppose that H is the only subgroup of G of order | H | and that |
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Unformatted text preview: G | &lt; . Prove that H must be normal. (6) Suppose that N and M are normal subgroups of G and that N T M = { e } . Prove that for any n N and m M , nm = mn , and hence that MN = M N as groups. (7) Give an example of three groups, K H G such that K is normal in H and H is normal in G , but K is not normal in G . (8) Prove that if | G | = p 2 for p a prime, then G is abelian. HINT: Try to show that Z ( G ) = G and use Cauchys and Lagranges theorems repeatedly. 1...
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