# Criteria for Groups and Burnside's Lemma

Let $G$ be a set of elements and $*$ a binary operation defined on the elements of $G$. Then $G$ is a group with respect to $*$ if the following holds:

• $a*(b*c) = (a*b)*c$ for all $a,b,c \in G$.
• There exists a left identity in $G$.
• For each $a \in G$, there exists a left inverse.

We could replace "left" with "right." Is there any relationship between this formulation and Burnside's Lemma? In other words, does the "left" or "right" criteria for groups come in handy when dealing with Burnside's Lemma (e.g could the number of orbits be the "left" part and the number of fixed points be the "right" part)?

• I'm not sure I understand the question. What connection is there between left and right and Burnside's lemma? – Qiaochu Yuan Dec 22 '10 at 2:56
• This was motivated from the following: mathoverflow.net/questions/50033/… – PrimeNumber Dec 22 '10 at 3:02
• E.g. are the the notions of "left" and "right" used in the context above the same as that in the MO question? – PrimeNumber Dec 22 '10 at 3:02
• Where are the notions of "left" and "right" used in that MO question? – Qiaochu Yuan Dec 22 '10 at 3:05
• Trevor, I think you have just proven the following Theorem by providing a counterexample: it is not true that any two questions that contain the words "left" and "right" are connected in any non-superficial way. – Alex B. Dec 22 '10 at 5:08

## 1 Answer

First, I assume you know that the two definitions yield objects which are not just isomorphic, but identical: from the 'left' definition it follows that the 'left' identity is also a 'right' identity and ergo it is 'the' identity, and the 'left' inverse is also a 'right' inverse and ergo 'the' inverse; and vice versa follows for the 'right' definition.

So I would say that the formulation of the axioms has little to do with Burnside's lemma per se; and instead is intended to provide a minimum set of requirements: only one side, left or right, is required to deduce that the resulting entities (identity, inverse) are actually /both-sided/. This is in contrast to other types of structures; e.g., modules, where a left module is not necessarily a right module.

Also, historically, the notation for the application of a group operation has varied between right to left and left to right over time. For example in W. R. Scott's "Group Theory" (1964), assuming x in S and g,h in G where G acts on S, the author writes "xhg" for what would now more typically would be written as "ghx". This has other terminological effects such as the precise meaning of "left coset" or "right coset", but of course the essential theory is the same.