# Mal'tsev product

From Encyclopedia of Mathematics

An operation on the class of all groups (denoted by ), hereditary on passing to subgroups of the factors; that is, if

and if a subgroup is chosen in each factor , then the subgroups , , generate a subgroup of which is the same as the product of the :

Direct sums and free products of groups are Mal'tsev products. There exist other Mal'tsev products, but Mal'tsev's problem on the existence of Mal'tsev products (other than direct or free) satisfying the associative law and certain other natural conditions is still (1989) open. (The Mal'tsev product originated in connection with this problem.)

#### References

[1] | A.G. Kurosh, "The theory of groups" , 1 , Chelsea (1955) (Translated from Russian) |

#### Comments

#### References

[a1] | O.N. Golovin, M.A. Bronshtein, "An axiomatic classification of exact operations" , Selected problems in algebra and logic , Novosibirsk (1973) pp. 40–96 (In Russian) |

**How to Cite This Entry:**

Mal'tsev product.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Mal%27tsev_product&oldid=12842

This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article