Prove that if is a prime and , then and . Deduce that has maximal subgroups.

Write . The maximal subgroups of have order . Note that and are thus maximal, so that . 34 more words

Prove that if is a prime and , then and . Deduce that has maximal subgroups.

Write . The maximal subgroups of have order . Note that and are thus maximal, so that . 34 more words

Let be a prime and let be a finite -group.

- Prove that is an elementary abelian -group.
- Prove that if is a normal subgroup such that is elementary abelian then . 980 more words

Let be a finite group. Prove that is nilpotent.

We will use Frattini’s Argument to show that every Sylow subgroup is normal.

Let be a Sylow subgroup. 36 more words

An element of a group is called a *nongenerator* if for all proper subgroups , is also proper. Prove that is the set of all nongenerators in . 107 more words

When is a finite group prove that if is normal, then . Give an explicit example when this containment does not hold if is not normal in . 217 more words

Compute , , , , and .

We begin with some lemmas.

Lemma 1: If is a subgroup of order 12, then . Proof: By Cauchy, contains an element of order 2 and an element of order 3. 215 more words

Let be a group and denote by the Frattini subgroup of . Prove that is characteristic.

We begin with some lemmas. Let denote the set of subgroups of ; recall that acts on by application: . 97 more words