Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Subgroup"

m (→‎Examples: Uncapitalized the N)
m (tinkering)
Line 1: Line 1:
 
{{stub}}
 
{{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 <math>H</math> is a subgroup of <math>G</math> if the elements of <math>H</math> constitute a subset of the set of elements  of <math>G</math>, and the group law on <math>H</math> agrees with group law on <math>G</math> where both are defined.  We may also write <math>H \subseteq G</math> or <math>H \le G</math>.
+
A '''subgroup''' is a [[group]] contained in another.  Specifically, let <math>H</math> and <math>G</math> be groups.  We say that <math>H</math> is a subgroup of <math>G</math> if the [[element]]s of <math>H</math> are a [[subset]] of the [[set]] of elements  of <math>G</math> and the group law on <math>H</math> agrees with group law on <math>G</math> where both are defined.  We may denote this by <math>H \subseteq G</math> or <math>H \le G</math>.
  
 
We say that <math>H</math> is a ''proper subgroup'' of <math>G</math> if <math>H \neq G</math>.
 
We say that <math>H</math> is a ''proper subgroup'' of <math>G</math> if <math>H \neq G</math>.
Line 14: Line 14:
 
2&2&3&0&1 \\
 
2&2&3&0&1 \\
 
3&3&0&1&2 \end{array}</cmath>
 
3&3&0&1&2 \end{array}</cmath>
−
there are three subgroups :  the group itself, <math>\{ 0 \}</math>, and the group <math>2 \mathbb{Z}/4\mathbb{Z}</math>, shown below.  This last subgroup is [[isomorphic]] to the additive group <math>\mathbb{Z}/2\mathbb{Z}</math>.
+
there are three subgroups :  the group itself, <math>\{ 0 \}</math>, and the group <math>2 \mathbb{Z}/4\mathbb{Z} = \{0, 2\}</math>, shown below.  This last subgroup is [[isomorphic]] to the additive group <math>\mathbb{Z}/2\mathbb{Z}</math>.
 
<cmath> \begin{array}{c|cc} & 0& 2 \\\hline
 
<cmath> \begin{array}{c|cc} & 0& 2 \\\hline
 
0&0&2 \\ 2&2&0 \end{array} </cmath>
 
0&0&2 \\ 2&2&0 \end{array} </cmath>
  
−
Every group is the largest subgroup of itself.  In a group with identity <math>e</math>, <math>\{e\}</math> is the smallest subgroup.
+
Every group is the largest subgroup of itself.  The set consisting of the [[identity]] element of a group is the smallest subgroup of that group.
  
−
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>.
  
−
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>.
+
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 15:01, 20 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. We say that $H$ is a subgroup of $G$ if the elements of $H$ are 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 denote this by $H \subseteq G$ 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 $2 \mathbb{Z}/4\mathbb{Z} = \{0, 2\}$, 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. The set consisting of the identity element of a group is the smallest subgroup of that group.

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Subgroup&oldid=23521"