When is the Number of p-Subgroups of a Group Satisfying a Property Congruent to 1 (mod p)?

Authors

Jason Fulman, Rose-Hulman Institute of Technology
Jeff Vanderkam, Rose-Hulman Institute of Technology

Document Type

Article

Publication Date

2-1993

First Advisor

Gary Sherman

Abstract

Let T be a property which holds for a group independent of whether or not this group is embedded in a group G or in a p-Sylow subgroup of G. Using a generalization of Sylow's second Theorem, we prove that if for any p-group P the number of subgroups of P satisfying T is congruent to 1 (mod p), then for any group G, the number of p-subgroups satisfying T is also congruent to 1 (mod p). As an application, we give simple proofs of several theorems, including the well-known Frobenius theorem.

Comments

MSTR 93-01

Recommended Citation

Fulman, Jason and Vanderkam, Jeff, "When is the Number of p-Subgroups of a Group Satisfying a Property Congruent to 1 (mod p)?" (1993). Mathematical Sciences Technical Reports (MSTR). 85.
https://scholar.rose-hulman.edu/math_mstr/85