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

Ammu Elizabeth Antony; Patali Komma; Viji Zachariah Thomas

doi:10.1007/s11856-021-2264-4

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

Ammu Elizabeth Antony, Patali Komma, Viji Zachariah Thomas

Research output: Contribution to journal › Article › peer-review

Abstract

In a p-group G of nilpotency class at most p +1, we prove that the exponent of the commutator subgroup γ₂(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.

Original language	English
Pages (from-to)	251-267
Number of pages	17
Journal	Israel Journal of Mathematics
Volume	247
Issue number	1
DOIs	https://doi.org/10.1007/s11856-021-2264-4
State	Published - Apr 2022
Externally published	Yes

Bibliographical note

Funding Information:
V. Z. Thomas acknowledges research support from SERB, DST, Government of India grant MTR/2020/000483. We sincerely thank the referee for helping us in greatly improving the exposition of this manuscript. We also thank Moravec who informed the third author that for groups of nilpotency class 5 computer evidence showed that exp(M(G)) ∣ (exp(G))2.

Publisher Copyright:
© 2021, The Hebrew University of Jerusalem.

ASJC Scopus subject areas

Mathematics (all)

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10.1007/s11856-021-2264-4

Cite this

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title = "A property of p-groups of nilpotency class p + 1 related to a theorem of Schur",

abstract = "In a p-group G of nilpotency class at most p +1, we prove that the exponent of the commutator subgroup γ2(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.",

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AU - Thomas, Viji Zachariah

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N2 - In a p-group G of nilpotency class at most p +1, we prove that the exponent of the commutator subgroup γ2(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.

AB - In a p-group G of nilpotency class at most p +1, we prove that the exponent of the commutator subgroup γ2(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.

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

Abstract

Bibliographical note

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