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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 nilpotent ideal. (Note that this ring may be noncommutative.) [With a hint from this forum post and… more →
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Review of Group Theory: Group Actions (Pt. IV Conjugation and the Class Equation Pt. II)
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wrote 2 years ago: 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
wrote 2 years ago: 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
wrote 2 years ago: 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
wrote 2 years ago: 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
wrote 2 years ago: 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
wrote 2 years ago: 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
wrote 2 years ago: 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
wrote 2 years ago: 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)
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wrote 2 years ago: 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
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wrote 2 years ago: Let be an odd prime and a group of order . Prove that the -th power map is a homomorphism and that . … more →
In a nonabelian p-group of order p³, the commutator subgroup and center are equal
wrote 2 years ago: 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
wrote 3 years ago: 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
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wrote 3 years ago: Let be a prime and a positive integer. Prove that if is a group of order then every subgroup of inde … more →
