Problems to submit for grading
Please organize your work, justify your steps, and write in complete sentences with correct punctuation. Once you finish, please follow the directions in the Canvas assignment to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!
Problem. Let $G$ be a finite group and $p$ a prime. Suppose $N\trianglelefteq G$. Prove that if $G/N$ has an element of order $p$, then $G$ has an element of order $p$.
Hint: Suppose \(xN\) has order $p$ in the group \(G/N\)…what can you deduce about the order of \(x\) in \(G\)? Can you find an element of order $p$ in \(\langle x \rangle\)?
We have not covered Cauchy’s Theorem, so please do not use it.
Problem. Let \(G\) be a group of order 24. Suppose that \(G\) has an abelian subgroup \(H\) of order \(8\). Prove that \(G\) has a normal subgroup \(N\) of order 4 or 8.
Hint: If \(H\) is not normal in \(G\), then \(H \neq gHg^{-1}\) for some \(g\in G\). If you set \(K = gHg^{-1}\), what can you say about \(\vert H\cap K \vert\)? What about \(\vert N_G(H\cap K) \vert\)?
Problem. Let $G$ be a group and $H\le G$.
- Prove that \(N_G(H)/C_G(H)\) is isomorphic to a subgroup of \(\operatorname{Aut}(H)\).
- Suppose that \(H\) is an abelian, normal subgroup of \(G\) and \(\gcd(\vert\operatorname{Aut}(H) \vert, \vert G:H \vert) = 1\). Prove that \(H \le Z(G)\).
Hint: For the first part, consider the function \(f:N_G(H) \rightarrow \operatorname{Aut}(H)\) defined by \(f(g) = \gamma_g\) where \(\gamma_g\) is the automorphism of \(H\) defined by \(\gamma_g(h)=ghg^{-1}\). Try to use the First Isomorphism Theorem. To do that, you’ll have to prove that for all \(a,b\in N_G(H)\), the function \(\gamma_{ab}\) is equal to the composition \(\gamma_{a}\circ \gamma_{b}\). For the second part, use the first; you have that \(G = N_G(H)\) and are trying to prove that \(G = C_G(H)\).
Problems for extra practice
Please try to solve some of these problems, but you do not turn them in. I’m happy to talk about these problems (or any others) if you have questions.
Problem. Let $G$ be a group and \(N\trianglelefteq G\). Prove that if \(N\) is a finite cyclic group, then every subgroup of \(N\) is normal in \(G\). (Can you also prove this when \(N\) is infinite cyclic?)
Problem. Let $G$ be a group. Suppose that $H\le G$, $N\trianglelefteq G$, and $HN = G$. Prove that if $H$ and $N$ are $p$-groups1, then $G$ is a $p$-group.
Problem. Use the First Isomorphism Theorem to prove each of the following group isomorphisms.
- For $F$ a field, $\operatorname{GL}_n(F) / \operatorname{SL}_n(F) \cong F^\times$.
- For $n=md$, $D_{2n}/\langle r^m \rangle \cong D_{2m}$.
Remember that $F^\times$ denotes \(F \setminus\{0\}\) as a group with respect to multiplication. To prove $G/N \cong H$, find a surjective homomorphism from $G$ to $H$ with kernel $N$.
Problems. Work some problems from D&F Section 3.2. Here are some to consider: 19, 21
Footnotes
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Definition. Let $p$ be a prime. A (possibly infinite) group $P$ is called a $p$-group if the order of every element of $P$ is a power of $p$; that is, $P$ is a $p$-group if for all $x\in P$, $|x| = p^k$ for some $k\in \mathbb{N}$. (This differs from the definition in Dummit and Foote, but we will later prove that the two definitions are equivalent for finite groups.) ↩