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We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.

We survey the current state of knowledge of bounds in the restricted Burnside problem. We make two conjectures which are related to the theory of PI-algebras.

We show that if $G$ is a group of exponent 5, and if $G$ satisfies the Engel-4 identity $[x,y,y,y,y]=1$, then $G$ is locally finite. By a result of Traustason, this implies that Engel-4 5-groups are locally finite. We also show that a group of exponent 5 is locally finite if and only if it satsifies the identity

$$[x,[y,z,z,z,z],[y,z,z,z,z]]=1.$$

This result implies that a group of exponent 5 is locally finite if its three generator subgroups are finite.

1991 Mathematics Subject Classification: 20D15, 20F45, 20F50.

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Yet Another Solution to the Burnside Problem for Matrix Semigroups

Bounds in the restricted Burnside problem

Engel-4 groups of exponent 5

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3 results

Yet Another Solution to the Burnside Problem for Matrix Semigroups

Bounds in the restricted Burnside problem

Engel-4 groups of exponent 5