Question

definition of group?

Answer 1

A group \(\langle G,\cdot\rangle\) is a set \(G\) along with a binary operator \(\cdot\) on \(G\), such that the following are true.

i) For all \(a,b\in G\) it is true that \(a\cdot b\) is in \(G\) (this is actually part of the definition of a binary operator).

ii) There exists an element \(e\in G\) such that \(a\cdot e=a=e\cdot a\) for all \(a\in G\) (so there is an element like 0). This element is called the identity element.

iii) \(\cdot\) is an associative operation. i.e. \((a\cdot b)\cdot c=a\cdot (b\cdot c)\) for all \(a,b,c\in G\).

iv) For all \(a\in G\) there exists an element \(b\in G\) such that \(a\cdot b=e=b\cdot a\). This element \(b\) is the inverse of \(a\) and we call it \(a^{-1}\).