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Difference between revisions of "Quotient group"

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''of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076880/q0768801.png" /> by a normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076880/q0768802.png" />''
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''of a group $G$ by a normal subgroup $N$
  
 
The group formed by the cosets (cf. [[Coset in a group|Coset in a group]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076880/q0768803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076880/q0768804.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076880/q0768805.png" />; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076880/q0768806.png" /> (cf. [[Normal subgroup|Normal subgroup]]). Multiplication of cosets is performed according to the formula
 
The group formed by the cosets (cf. [[Coset in a group|Coset in a group]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076880/q0768803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076880/q0768804.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076880/q0768805.png" />; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076880/q0768806.png" /> (cf. [[Normal subgroup|Normal subgroup]]). Multiplication of cosets is performed according to the formula

Revision as of 03:02, 20 June 2012

of a group $G$ by a normal subgroup $N$

The group formed by the cosets (cf. Coset in a group) , , of ; it is denoted by (cf. Normal subgroup). Multiplication of cosets is performed according to the formula

The unit of the quotient group is the coset , and the inverse of the coset is .

The mapping is a group epimorphism of onto , called the canonical epimorphism or natural epimorphism. If is an arbitrary epimorphism of onto a group , then the kernel of is a normal subgroup of , and the quotient group is isomorphic to ; more precisely, there is an isomorphism of onto such that the diagram

is commutative, where is the natural epimorphism .

A quotient group of a group can be defined, starting from some congruence on (cf. Congruence (in algebra)), as the set of classes of congruent elements relative to multiplication of classes. All possible congruences on a group are in one-to-one correspondence with its normal subgroups, and the quotient groups by the congruences are the same as those by the normal subgroups. A quotient group is a normal quotient object in the category of groups.


Comments

References

[a1] P.M. Cohn, "Algebra" , I , Wiley (1982) pp. Sect. 9.1
How to Cite This Entry:
Quotient group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quotient_group&oldid=16178
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article