On p -Group Isomorphism: search-to-decision, counting-to-decision, and nilpotency class reductions via tensors

In this paper we study some classical complexity-theoretic questions regarding G roup I somorphism (G p I). We focus on p -groups (groups of prime power order) with odd p , which are believed to be a bottleneck case for G p I, and work in the model of matrix groups over finite fields. Our main results are as follows. • Although search-to-decision and counting-to-decision reductions have been known for over four decades for G raph I somorphism (GI), they had remained open for G p I, explicitly asked by Arvind & Torán (Bull. EATCS, 2005). Extending methods from T ensor I somorphism (Grochow & Qiao, ITCS 2021), we show moderately exponential-time such reductions within p -groups of class 2 and exponent p . • D espite the widely held belief that p -groups of class 2 and exponent p are the hardest cases of GpI, there was no reduction to these groups from any larger class of groups. Again using methods from Tensor Isomorphism (ibid.), we show the first such reduction, namely from isomorphism testing of p -groups of “small” class and exponent p to those of class two and exponent p . For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of GI . Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets. The analysis of one of our gadgets relies on a classical result from group theory regarding random generation of classical groups (Kantor & Lubotzky, Geom. Dedicata, 1990). For the nilpotency class reduction, we combine a runtime analysis of the Lazard correspondence with T ensor I somorphism -completeness results (Grochow & Qiao, ibid.).