#### Review of Group Theory: Group Actions (Pt. IV Conjugation and the Class Equation Pt. II) — 5 comments

: Point of post: This is a continuation of this post. From this we can derive some interesting results … more →

#### Prove that the augmentation ideal of a given group ring is nilpotent

: Let be a prime and let be a finite -group. Prove that the augmentation ideal in the group ring is a … more →

#### Compute the nilradical of a given group ring

: Let be a ring with . Let be a prime and let be an abelian group of order . Prove that the nilradical … more →

#### The normalizer of a maximal Sylow intersection is not a p-subgroup

: Suppose that over all pairs of Sylow -subgroups, and are chosen so that is maximal. Prove that is no … more →

#### Compute the Frattini subgroup and number of maximal subgroups of a nonabelian group of order p³

: Prove that if is a prime and is a nonabelian group of order then and . Deduce that has maximal subgr … more →

#### Count the maximal subgroups of a finite p-group

: Let be a prime, let be a finite -group, and let be elementary abelian of rank . Prove that has exact … more →

#### Some properties of nonabelian p groups of order p³

: Prove that if is a prime and a nonabelian group of order , then and . By Lagrange, there are 4 possi … more →

#### For odd primes p, a p-group whose every subgroup is normal is abelian

: Let be an odd prime and let be a -group. Prove that if every subgroup of is normal then is abelian. … more →

#### With p an odd prime, every noncyclic p-group contains a normal direct product of two copies of Cyc(p) — 6 comments

: Let be an odd prime. Prove that if is a noncyclic -group then contains a normal subgroup with . Dedu … more →

#### Properties of the p-power map on a group whose order is the cube of an odd prime — 6 comments

: Let be an odd prime and a group of order . Prove that the -th power map is a homomorphism and that . … more →

Tags: aadf, center (group), power map

#### In a nonabelian p-group of order p³, the commutator subgroup and center are equal

: Prove that if is a prime and a nonabelian group of order , then . Since is nonabelian, we have . Mor … more →

#### A p-group contains subgroups of every order allowed by Lagrange's Theorem

: Let be a prime and let be a group of order . Prove that has a subgroup of order for all . [Hint: Use … more →

#### In a p-group, every proper subgroup of minimal index is normal — 8 comments

: Let be a prime and a positive integer. Prove that if is a group of order then every subgroup of inde … more →

All →