Schur's exponent conjecture -- counterexamples of exponent 5 and exponent 9

2020/08/16 by Vaughan-Lee, Michael
#20D15 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2008.06848

Abstract

There is a long-standing conjecture attributed to I Schur that if G is a finite group with Schur multiplier M(G) then the exponent of M(G) divides the exponent of G. It is easy to see that this conjecture holds for exponent 2 and exponent 3, but it has been known since 1974 that the conjecture fails for exponent 4. In this note I give an example of a group G with exponent 5 with Schur multiplier M(G) of exponent 25, and an example of a group A of exponent 9 with Schur multiplier M(A) of exponent 27.

Related

• A lemma on the exponent of Schur multiplier of p groups with good power structure
• On the Exponent of the Schur Multiplier
• On the exponent conjecture of Schur
• Schur's exponent conjecture II
• Exponents of Bogomolov multipliers