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A non-empty subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s0908601.png" /> of a [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s0908602.png" /> which itself is a group with respect to the operation defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s0908603.png" />. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s0908604.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s0908605.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s0908606.png" /> if and only if: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s0908607.png" /> contains the product of any two elements from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s0908608.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s0908609.png" /> contains together with any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086010.png" /> the inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086011.png" />. In the cases of finite and periodic groups, condition 2) is superfluous.
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A non-empty subset $H$ of a [[Group|group]] $G$ which itself is a group with respect to the operation defined on $G$. A subset $H$ of a group $G$ is a subgroup of $G$ if and only if: 1) $H$ contains the product of any two elements from $H$; and 2) $H$ contains together with any element $h$ the inverse $h^{-1}$. In the cases of finite and periodic groups, condition 2) is superfluous.
  
The subset of a group G consisting of the element 1 only is clearly a subgroup; it is called the unit subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086012.png" /> and is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086013.png" />. Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086014.png" /> itself is a subgroup. A subgroup different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086015.png" /> is called a proper subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086016.png" />. A proper subgroup of an infinite group can be isomorphic to the group itself. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086017.png" /> itself and the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086018.png" /> are called improper subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086019.png" />, while all the others are called proper ones.
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The subset of a group G consisting of the element 1 only is clearly a subgroup; it is called the unit subgroup of $G$ and is usually denoted by $E$. Also, $G$ itself is a subgroup. A subgroup different from $G$ is called a proper subgroup of $G$. A proper subgroup of an infinite group can be isomorphic to the group itself. The group $G$ itself and the subgroup $E$ are called improper subgroups of $G$, while all the others are called proper ones.
  
The set-theoretic intersection of any two (or any set of) subgroups of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086020.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086021.png" />. The intersection of all subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086022.png" /> containing all elements of a certain non-empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086023.png" /> is called the subgroup generated by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086024.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086026.png" /> consists of one element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086028.png" /> is called the cyclic subgroup of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086029.png" />. A group that coincides with one of its cyclic subgroups is called a [[Cyclic group|cyclic group]].
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The set-theoretic intersection of any two (or any set of) subgroups of a group $G$ is a subgroup of $G$. The intersection of all subgroups of $G$ containing all elements of a certain non-empty set $M$ is called the subgroup generated by the set $M$ and is denoted by $\{M\}$. If $M$ consists of one element $a$, then $\{a\}$ is called the cyclic subgroup of the element $a$. A group that coincides with one of its cyclic subgroups is called a [[Cyclic group|cyclic group]].
  
A set-theoretic union of subgroups is, in general, not a subgroup. By the join of subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086031.png" />, one means the subgroup generated by the union of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086032.png" />.
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A set-theoretic union of subgroups is, in general, not a subgroup. By the join of subgroups $H_i$, $i\in I$, one means the subgroup generated by the union of the sets $H_i$.
  
The product of two subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086034.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086035.png" /> is the set consisting of all possible (different) products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086038.png" />. The product of two subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086039.png" /> is a subgroup if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086040.png" />, and in that case the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086041.png" /> coincides with the subgroup generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086043.png" /> (i.e. with the join of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086045.png" />).
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The product of two subsets $S_1$ and $S_2$ of a group $G$ is the set consisting of all possible (different) products $s_1s_2$, where $s_1\in S_1$, $s_2\in S_2$. The product of two subgroups $H_1,H_2$ is a subgroup if and only if $H_1H_2=H_2H_1$, and in that case the product $H_1H_2$ coincides with the subgroup generated by $H_1$ and $H_2$ (i.e. with the join of $H_1$ and $H_2$).
  
A homomorphic image of a subgroup is a subgroup. If a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086046.png" /> is isomorphic to a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086047.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086048.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086049.png" /> can be imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090860/s09086050.png" /> (as groups). If one is given two groups and each of them is isomorphic to a proper subgroup of the other, it does not necessarily follow that these groups themselves are isomorphic (cf. also [[Homomorphism|Homomorphism]]; [[Isomorphism|Isomorphism]]).
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A homomorphic image of a subgroup is a subgroup. If a group $G_1$ is isomorphic to a subgroup $H$ of a group $G$, one says that $G_1$ can be imbedded in $G$ (as groups). If one is given two groups and each of them is isomorphic to a proper subgroup of the other, it does not necessarily follow that these groups themselves are isomorphic (cf. also [[Homomorphism|Homomorphism]]; [[Isomorphism|Isomorphism]]).
  
  

Latest revision as of 14:22, 30 August 2014

A non-empty subset $H$ of a group $G$ which itself is a group with respect to the operation defined on $G$. A subset $H$ of a group $G$ is a subgroup of $G$ if and only if: 1) $H$ contains the product of any two elements from $H$; and 2) $H$ contains together with any element $h$ the inverse $h^{-1}$. In the cases of finite and periodic groups, condition 2) is superfluous.

The subset of a group G consisting of the element 1 only is clearly a subgroup; it is called the unit subgroup of $G$ and is usually denoted by $E$. Also, $G$ itself is a subgroup. A subgroup different from $G$ is called a proper subgroup of $G$. A proper subgroup of an infinite group can be isomorphic to the group itself. The group $G$ itself and the subgroup $E$ are called improper subgroups of $G$, while all the others are called proper ones.

The set-theoretic intersection of any two (or any set of) subgroups of a group $G$ is a subgroup of $G$. The intersection of all subgroups of $G$ containing all elements of a certain non-empty set $M$ is called the subgroup generated by the set $M$ and is denoted by $\{M\}$. If $M$ consists of one element $a$, then $\{a\}$ is called the cyclic subgroup of the element $a$. A group that coincides with one of its cyclic subgroups is called a cyclic group.

A set-theoretic union of subgroups is, in general, not a subgroup. By the join of subgroups $H_i$, $i\in I$, one means the subgroup generated by the union of the sets $H_i$.

The product of two subsets $S_1$ and $S_2$ of a group $G$ is the set consisting of all possible (different) products $s_1s_2$, where $s_1\in S_1$, $s_2\in S_2$. The product of two subgroups $H_1,H_2$ is a subgroup if and only if $H_1H_2=H_2H_1$, and in that case the product $H_1H_2$ coincides with the subgroup generated by $H_1$ and $H_2$ (i.e. with the join of $H_1$ and $H_2$).

A homomorphic image of a subgroup is a subgroup. If a group $G_1$ is isomorphic to a subgroup $H$ of a group $G$, one says that $G_1$ can be imbedded in $G$ (as groups). If one is given two groups and each of them is isomorphic to a proper subgroup of the other, it does not necessarily follow that these groups themselves are isomorphic (cf. also Homomorphism; Isomorphism).


Comments

References

[a1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[a2] M. Hall jr., "The theory of groups" , Macmillan (1959) pp. 124
How to Cite This Entry:
Subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subgroup&oldid=33209
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article