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.