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. Assume $G$ acts on $X$.
- Let \(a\in X\) and \(g\in G\). Prove that \(gG_ag^{-1} = G_{g\cdot a}\).
- Let \(a\in X\). Prove that if the action is transitive and faithful, then \(\bigcap_{g\in G} gG_ag^{-1} = 1\).
- Prove that if \(G\) is abelian and the action is transitive and faithful, then \(\vert G\vert = \vert X\vert\).
Problem. Let \(G\) be a group of order 108. Suppose that \(G\) has a subgroup \(P\) of order \(27\). Prove that \(G\) has a normal subgroup of order 9 or 27.
Hint: Act on $G/P$ by left multiplication.
Problem. Let $G$ be a finite group. Prove that if $G$ has exactly $3$ conjugacy classes of elements, then \(\vert G\vert \le 6\).
Hint: Think about the class equation. (In other words, remember that that conjugacy classes partition $G$, so \(\vert G\vert\) is equal to the sum of the sizes of the conjugacy classes, each of which is a divisor of \(\vert G\vert\).) There are various approaches at this point, but consider dividing the the equation by \(\vert G\vert\) and then analyzing the possibilities.
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 an infinite group. Prove that $G$ has a subgroup of finite index, then it has a normal subgroup of finite index.
Problem. Let \(G\) be a group of odd order. Prove that for all nontrivial \(a\in G\), \(a\) and \(a^{-1}\) are not conjugate.
Hint: Suppose $a$ and $a^{-1}$ are conjugate. Then there exists some $g\in G$ such that $gag^{-1} = a^{-1}$. What can you say about $g^2$?
Problems. Work some problems from D&F Section 1.7. Here are some to consider: 8, 9, 11, 12
Problems. Work some problems from D&F Section 4.1. Here are some to consider: 4, 10
Problems. Work some problems from D&F Section 4.2. Here are some to consider: 4, 9
- To clarify #4, you want to use the left regular representation of \(Q_8\) (i.e. the action of \(Q_8\) on itself by left multiplication) to explicitly find two elements \(a,b\in S_8\) such that \(\langle a, b \rangle \cong Q_8\).
Problems. Work some problems from D&F Section 4.3. Here are some to consider: 2, 5