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This tag is for questions relating to "Quotient Group".

A quotient group or factor group is a group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.

Definition: If $G$ is a group and $N$ is a normal subgroup of group $G$, then the set $G/N$ of all cosets of $N$ in $G$ is a group with respect to the multiplication of cosets. It is called the quotient group or factor group of $G$ by $N$. The identity element of the quotient group $G/N$ by $N$.

  • Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup.
  • In category theory, quotient groups are examples of quotient objects, which are dual to subobjects.
  • The quotient group construction can be viewed as a generalization of modular arithmetic to arbitrary groups. In fact, the quotient group $G/N$ is read "$G$ mod $N.$"
  • It can be verified that the set of self-conjugate elements of $G$ forms an abelian group $Z$ which is called the center of $G$.