  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.

Group Theory