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Problem Set 3

Instructions: This set is due October 16th.

Problem 1: Let $H$ and $K$ be groups and let $\phi: K\to \mathrm{Aut}(H)$ be a homomorphism.

a. Given an automorphism $f: K\to K$, the map $\phi\circ f$ is a homomorphism from $K$ to $\mathrm{Aut}(H)$. Show that the groups $H\rtimes_{\phi}K$ and $H\rtimes_{\phi\circ f}K$ are isomorphic.

b. Suppose $f:H\to H$ is an automorphism. If $\sigma:H \to H$ is an automorphism, so is the conjugate $f\circ \sigma\circ f^{-1}$. Show that $\gamma_{f}(\sigma)=f\circ\sigma\circ f^{-1}$ is an automorphism of $\mathrm{Aut}(H)$, and prove that $H\rtimes_{\phi}K$ and $H\rtimes_{\gamma_{f}\circ\phi}K$ are isomorphic.

Problem 2: Construct a non-abelian group of order $75$ and (using the previous problem) show that it is unique up to isomorphism. Is it always the case that there is only one nonabelian group of order $pq^2$ when $q>p$?

Problem 3: DF p. 231–232 Problems 15, 21 and 22 about Boolean rings.

Problem 4: DF p. 250, problem 34. Note that $R$ is not assumed commutative, and “ideal” means two-sided ideal.

Problem 5: Classify the following ideals in $\Z[x]$ as prime but not maximal, maximal, or not prime. (Hint: look at the quotient rings).

a. $(10)=10\Z[x]$. b. $(5, x+7)$ c. $(x^2-3)$ d. $\lbrace f\in\Z[x] : f(2)=0\rbrace$