Minimal embeddings of small finite groups

Document Type : Research Paper


1 Department of Mathematics, Cork Institute of Technology, Bishopstown, Cork, Ireland.

2 Department of Computing and Mathematics, Waterford Institute of Technology, Waterford, Ireland


We determine the groups of minimal order in which all groups of order $n$ can embedded for $ 1 \le n \le 15$‎. ‎We further determine the minimal order of a group in which all groups of order $n$ or less can be embedded‎, ‎also for $ 1 \le n \le 15$‎.


[1] A. Cayley, Desiderata and Suggestions, no. 1, The Theory of Groups, Amer. J. Math., I (1878) 50–52. Reprinted in
The Collected Mathematical Papers of Arthur Cayley, Cambridge University Press, x 1898.
[2] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Clarendon Press,
Oxford, 1985.
[3] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.2, 2013,
[4] R. Heffernan, D. MacHale and B. McCann, Cayley’s Theorem Revisited: Embeddings of Small Finite Groups, Math.
Mag., 91-2 (2018) 103–111.
[5] I. M. Isaacs, Finite Group Theory, Graduate Studies in Mathematics, 92, American Mathmeatical Society, Providence, Rhode Island, 2008.
[6] B. McCann, On products of cyclic and elementary abelian p-groups, Publ. Math. Debrecen, 91 (2017) 185–216.
[7] R. A. Wilson et al., ATLAS of Finite Group Representations - Version 3,
[8] R. A. Wilson, The Finite Simple Groups, Springer, London, 2009