# 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.