# Subgroup

A subgroup of a group $$(G,*)$$ is a group of the form $$(H,*)$$, where $$H \subset G$$. We usually say simply that $$H$$ is a subgroup of $$G$$.

For a subset of a group $$G$$ to be a subgroup, it needs to satisfy all of the group axioms itself: closure, associativity, identity, and inverse. We get associativity for free because $$G$$ is a group. So the requirements of a subgroup $$H$$ are:

1. Closure: For any $$x, y$$ in $$H$$, $$x*y$$ is in $$H$$.

2. Identity: The identity $$e$$ of $$G$$ is in $$H$$.

3. Inverses: For any $$x$$ in $$H$$, $$x^{-1}$$ is also in $$H$$.

A subgroup is called normal if it is closed under conjugation.

The subgroup relation is transitive: if $$H$$ is a subgroup of $$G$$, and $$I$$ is a subgroup of $$H$$, then $$I$$ is a subgroup of $$G$$.

# Examples

Any group is a subgroup of itself. The trivial group is a subgroup of every group.

For any integer $$n$$, the set of multiples of $$n$$ is a subgroup of the integers (under addition).

Parents:

• Group

The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.