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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:

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


$\displaystyle 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.

Robert G. Brown 2017-07-11