# Variations on a Theme: $A_4$ Definitely Has no Subgroup of Order Six!

by

Michael Brennan (Cork Institute of Technology Ireland) and Des Machale (University College Ireland)

This article originally appeared in:
Mathematics Magazine
February, 2000

Subject classification(s): Algebra and Number Theory | Abstract Algebra | Groups
Applicable Course(s): 4.2 Mod Algebra I & II

The authors present 11 different proofs that the alternating group$A_4$ has no subgroup of order 6, and hence that the converse of Lagrange's theorem is false.

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