On p-central groups

TL;DR

This paper extends the concept of free p-central groups to include p=2, introduces a new series, and calculates Schur multipliers, revealing their properties and relationships to the group's exponent.

Contribution

It generalizes free p-central groups to p=2 and provides methods to compute their Schur multipliers and analyze their exponents.

Findings

Calculated Schur multipliers for free p-central groups.

Proved the exponent of the Schur multiplier divides the group exponent.

Determined the exponential rank of p-central groups.

Abstract

We extend the notion of free $p$-central groups for odd primes $p$ to the case $p=2$ by introducing a variant of the lower $p$-central series. This enables us to calculate Schur multipliers of free $p$-central groups. We also prove that for any $p$-central group the exponent of its Schur multiplier divides the exponent of the group, and determine its exponential rank.