# Mal'tsev product

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

$$G=\prod_{i\in I}^\circ G_i$$

and if a subgroup $H_i$ is chosen in each factor $G_i$, then the subgroups $H_i$, $i\in I$, generate a subgroup $H$ of $G$which is the same as the product of the $H_i$:

$$H=\prod_{i\in I}^\circ H_i.$$

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=43552