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Difference between revisions of "Subgroup"

(New page: {{stub}} A '''subgroup''' is a group contained in another. Specifically, let <math>H</math> and <math>G</math> be groups (with group laws written multiplicatively). We say that <mat...)
 
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In a group <math>G</math>, the intersection of a family of subgroups of <math>G</math> is a subgroup of <math>G</math>.  Thus for any collection <math>X</math> of elements of <math>G</math>, there exists a smallest subgroup containing these elements.  This is called the subgroup generated by <math>X</math>.
 
In a group <math>G</math>, the intersection of a family of subgroups of <math>G</math> is a subgroup of <math>G</math>.  Thus for any collection <math>X</math> of elements of <math>G</math>, there exists a smallest subgroup containing these elements.  This is called the subgroup generated by <math>X</math>.
  
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In the additive group <math>\mathbb{Z}</math>, all subgroups are of the form <math>n \mathbb{Z}</math>, for some integer <math>N</math>.  In particular, for <math>n=1</math>, we have the integers themselves, and for <math>n=0</math>, we have <math>\{0\}</math>.
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In the additive group <math>\mathbb{Z}</math>, all subgroups are of the form <math>n \mathbb{Z}</math>, for some integer <math>n</math>.  In particular, for <math>n=1</math>, we have the integers themselves, and for <math>n=0</math>, we have <math>\{0\}</math>.
  
 
== See Also ==
 
== See Also ==

Revision as of 14:17, 19 February 2008

This article is a stub. Help us out by expanding it.

A subgroup is a group contained in another. Specifically, let $H$ and $G$ be groups (with group laws written multiplicatively). We say that $H$ is a subgroup of $G$ if the elements of $H$ constitute a subset of the set of elements of $G$, and the group law on $H$ agrees with group law on $G$ where both are defined. We may also write $H \subseteq G$ or $H \le G$.

We say that $H$ is a proper subgroup of $G$ if $H \neq G$.

Examples

In the additive group $\mathbb{Z}/4\mathbb{Z}$, shown below, \[\begin{array}{c|cccc} &0&1&2&3 \\\hline 0&0&1&2&3 \\ 1&1&2&3&0 \\ 2&2&3&0&1 \\ 3&3&0&1&2 \end{array}\] there are three subgroups : the group itself, $\{ 0 \}$, and the group $2 \mathbb{Z}/4\mathbb{Z}$, shown below. This last subgroup is isomorphic to the additive group $\mathbb{Z}/2\mathbb{Z}$. \[\begin{array}{c|cc} & 0& 2 \\\hline 0&0&2 \\ 2&2&0 \end{array}\]

Every group is the largest subgroup of itself. In a group with identity $e$, $\{e\}$ is the smallest subgroup.

In a group $G$, the intersection of a family of subgroups of $G$ is a subgroup of $G$. Thus for any collection $X$ of elements of $G$, there exists a smallest subgroup containing these elements. This is called the subgroup generated by $X$.

In the additive group $\mathbb{Z}$, all subgroups are of the form $n \mathbb{Z}$, for some integer $n$. In particular, for $n=1$, we have the integers themselves, and for $n=0$, we have $\{0\}$.

See Also

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