The triple product property (TPP) for subsets of a finite group was introduced by Henry Cohn and Christopher Umans in 2003 as a tool for the study of the complexity of matrix multiplication. This note records some consequences of the simple observation that if (S1,S2,S3) is a TPP triple in a finite group G, then so is (dS1a,dS2b,dS3c) for any a,b,c,d∈G.
Let si:=∣Si∣ for 1≤i≤3. First we prove the inequality s1(s2+s3−1)≤∣G∣ and show some of its uses. Then we show (something a little more general than) that if G has an abelian subgroup of index v, then s1s2s3 ≤v2 ∣G∣.