<-- Go Back Last Updated: 12/01/2025

Subgroup

Given a group \((G,\cdot)\), a 'subgroup' of \(G\) is a non-empty subset \(H\subseteq G\) such that \((H,\cdot)\) is also a group, i.e. \(\forall g,h\in H: g\cdot h\in H,\: e_G\in H,\: g^{-1}\in H\). One writes \(H\leq G\), and calls \(H\) 'proper' if \(H\neq G\) and 'non-trivial' if \(H\neq \{e_G\}\).