A property of p-groups of nilpotency class p + 1 related to a theorem of Schur

In a p -group G of nilpotency class at most p +1, we prove that the exponent of the commutator subgroup γ 2 ( G ) divides the exponent of G/Z ( G ). As a consequence, we deduce that the exponent of the Schur multiplier divides the exponent of G for a p -group of nilpotency class at most p , odd orde...

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Bibliographic Details
Published in:Israel journal of mathematics 2022, Vol.247 (1), p.251-267
Main Authors: Antony, Ammu Elizabeth, Komma, Patali, Thomas, Viji Zachariah
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Commutators
Group Theory and Generalizations
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Subgroups
Theoretical
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Summary:In a p -group G of nilpotency class at most p +1, we prove that the exponent of the commutator subgroup γ 2 ( G ) divides the exponent of G/Z ( G ). As a consequence, we deduce that the exponent of the Schur multiplier divides the exponent of G for a p -group of nilpotency class at most p , odd order nilpotent groups of class at most 5, center-by-metabelian groups of exponent p , and a class of groups which includes p -groups of maximal class and potent p -groups.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-021-2264-4