Word maps with constants on symmetric groups

Abstract We study word maps with constants on symmetric groups. Even though there are non‐trivial mixed identities of bounded length that valid for all groups, we show no such can hold in the limit a metric sense. Moreover, prove and content, short enough, have an image positive diameter, measured normalized Hamming metric, which is from below terms length. Finally, also every self‐map finite non‐abelian simple group actually map G .