Computational Complexity of Finding Subgroups of a Given Order

TL;DR

Problem: Given a finite group described by its Cayley table, determine whether there exists a subgroup of a specified order $m$. Approach and results: the general problem is $NP$-complete via a Hamiltonian Cycle reduction to a subgroup problem in a constructed permutation group $G'=ig\langle \sigma_{ij}\mid (i,j)\in E\big\rangle$, while there is a linear-time algorithm in the abelian case when the input is the Cayley table, running in $\tilde{O}(k)$ time for group order $k$. Contributions: (i) $NP$-completeness for non-abelian groups; (ii) a $\tilde{O}(k)$-time abelian-group algorithm; (iii) discussion of how hardness transfers across representations and the abelian/non-abelian complexity gap. Significance: clarifies the impact of representation on computational difficulty in subgroup-finding problems and informs algorithm design in computational group theory.

Abstract

We study the problem of finding a subgroup of a given order in a finite group, where the group is represented by its Cayley table. We establish that this problem is NP-hard in the general case by providing a reduction from the Hamiltonian Cycle problem. Additionally, we analyze the complexity of the problem in the special case of abelian groups and present a linear-time algorithm for finding a subgroup of a given order when the input is given in the form of a Cayley table. To the best of our knowledge, no prior work has addressed the complexity of this problem under the Cayley table representation. Our results also provide insight into the computational difficulty of finding subgroup across different ways of groups representations.

Theorems & Definitions (11)

• Theorem 2.1
• proof
• Theorem 3.1
• Proposition 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.2
• proof