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

M. Vaughan-Lee
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引用次数: 5

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.
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舒尔指数猜想——指数5和指数9的反例。
有一个由I Schur提出的长期猜想,如果$G$是具有Schur乘子$M(G)$的有限群,则$M(G)$的指数除以$G$的指数。很容易看出,这个猜想对指数2和指数3成立,但自1974年以来,人们已经知道,这个猜想对指数4不成立。在本文中,我给出了一个指数为5的群$G$和指数为25的舒尔乘法器$M(G)$的例子,以及指数为9的群$ a $和指数为27的舒尔乘法器$M(a)$的例子。
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