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