Ribet's lemma

In mathematics, Ribet's lemma gives conditions for a subgroup of a product of groups to be the whole product group. It was introduced by Ribet (1976, lemma 5.2.2).

Statement

Suppose G₁×...×G_n is a product of perfect groups. Then any subgroup of this product that maps onto all the factors G_i for i=1, ..., n is the whole product group.

References

• Ribet, Kenneth A. (1976), "Galois action on division points of Abelian varieties with real multiplications", Amer. J. Math., 98 (3): 751–804, doi:10.2307/2373815, JSTOR 2373815, MR 0457455