Difference between revisions of "Quotient group"
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− | ''of a group | + | ''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 |
Quotient group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quotient_group&oldid=27032