Subgroups

A subgroup is a subset of elements in the group that is itself a group, for example the set of all real numbers less zero $\mathbb{R}^* = \mathbb{R}\backslash 0$ is a subgroup of $\mathbb{C}$ , and the set of all rational numbers (less zero) is similarly a subgroup of $\mathbb{R}^*$ . The mathamatical notation for a subgroup is the same as that of a subset:

$$SO(3) \subset O(3)$$

or

$$Z_1 \subset \mathbb{R}^*$$

The trivial group $Z_1$ is obviously a subgroup of all groups. Also a group is always its own subgroup. A simple group is one with only these two subgroups - one cannot find any set of elements smaller than the entire group except the trivial group that is a subset.