On a subgroup contained in some words with a bounded length (original) (raw)

1992, Discrete Mathematics

Hamidoune, Y.O., On a subgroup contained in some words with a bounded length, Discrete Mathematics 103 (1992) 171-176. Let G be a group and let A and B be two finite nonvoid subsets of G such that 1$ B. Using results of Kemperman, we show that either IA U B U ABl b IAl + IBI or there exists a nonnull subgroup contained in A U B U Al?. As an application we obtain the following result: Let A,, 4,. . , A, be subsets of a finite group G such that 1 $ A,; 2 c i s k and IA,1 + IA,1 +. . + jAkl 3 ICI. The union of sets of the form A,,A,,. .. Aij; 1 s i, <i, <.. . <ii s k must include a nonnull subgroup. In particular if B is a subset of G\l such that k IBI 2 ICI, the set B U /3* U.. . U B* must contain a nonnull subgroup.