Definition:Proper Subgroup

Let $$\left({G, \circ}\right)$$ be a group.

Then $$\left({H, \circ}\right)$$ is a proper subgroup of $$\left({G, \circ}\right)$$ iff:


 * $$\left({H, \circ}\right)$$ is a subgroup of $$\left({G, \circ}\right)$$;
 * $$H \ne G$$, i.e. $$H \subset G$$.

The notation $$H < G$$, or $$G > H$$, means "$$H$$ is a proper subgroup of $$G$$".

If $$H$$ is a subgroup of $$G$$, but it is not specified whether $$H = G$$ or not, then we write $$H \le G$$, or $$G \ge H$$.