# Groups: Simple and Solvable

## $$\star$$ Fall 2016 #7#algebra/qual/work

• Define what it means for a group $$G$$ to be solvable.

• Show that every group $$G$$ of order 36 is solvable.

Hint: you can use that $$S_4$$ is solvable.

## Spring 2015 #4#algebra/qual/work

Let $$N$$ be a positive integer, and let $$G$$ be a finite group of order $$N$$.

• Let $$\operatorname{Sym}^*G$$ be the set of all bijections from $$G\to G$$ viewed as a group under composition. Note that $$\operatorname{Sym}^*G \cong S_N$$. Prove that the Cayley map \begin{align*} C: G&\to \operatorname{Sym}^*G\\ g &\mapsto (x\mapsto gx) \end{align*} is an injective homomorphism.

• Let $$\Phi: \operatorname{Sym}^*G\to S_N$$ be an isomorphism. For $$a\in G$$ define $${\varepsilon}(a) \in \left\{{\pm 1}\right\}$$ to be the sign of the permutation $$\Phi(C(a))$$. Suppose that $$a$$ has order $$d$$. Prove that $${\varepsilon}(a) = -1 \iff d$$ is even and $$N/d$$ is odd.

• Suppose $$N> 2$$ and $$n\equiv 2 \operatorname{mod}4$$. Prove that $$G$$ is not simple.

Hint: use part (b).

## Spring 2014 #1#algebra/qual/work

Let $$p, n$$ be integers such that $$p$$ is prime and $$p$$ does not divide $$n$$. Find a real number $$k = k (p, n)$$ such that for every integer $$m\geq k$$, every group of order $$p^m n$$ is not simple.

## Fall 2013 #1#algebra/qual/work

Let $$p, q$$ be distinct primes.

• Let $$\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu \in {\mathbf{Z}}_p$$ be the class of $$q\operatorname{mod}p$$ and let $$k$$ denote the order of $$\mkern 1.5mu\overline{\mkern-1.5muq\mkern-1.5mu}\mkern 1.5mu$$ as an element of $${\mathbf{Z}}_p^{\times}$$. Prove that no group of order $$pq^k$$ is simple.

• Let $$G$$ be a group of order $$pq$$, and prove that $$G$$ is not simple.

## Spring 2013 #4#algebra/qual/work

Define a simple group. Prove that a group of order 56 can not be simple.

## Fall 2019 Midterm #3#algebra/qual/work

Show that there exist no simple groups of order 148.

#7 #algebra/qual/work #4 #1 #3