Editing Glossary of group theory (section)

== I ==
{{glossary}}
{{term|1=index of a subgroup}}
{{defn|1=The [[index of a subgroup|index]] of a {{gli|subgroup}} {{math|''H''}} of a group {{math|''G''}}, denoted {{math|{{abs|''G'' : ''H''}}}} or {{math|{{bracket|''G'' : ''H''}}}} or {{math|(''G'' : ''H'')}}, is the number of [[coset]]s of {{math|''H''}} in {{math|''G''}}. For a {{gli|normal subgroup}} {{math|''N''}} of a group {{math|''G''}}, the index of {{math|''N''}} in {{math|''G''}} is equal to the {{gli|order of a group|order}} of the {{gli|quotient group}} {{math|''G'' / ''N''}}. For a {{gli|finite group|finite}} subgroup {{math|''H''}} of a finite group {{math|''G''}}, the index of {{math|''H''}} in {{math|''G''}} is equal to the quotient of the orders of {{math|''G''}} and {{math|''H''}}.}}
{{term|1=isomorphism}}
{{defn|1=Given two groups {{math|(''G'', •)}} and {{math|(''H'', ·)}}, an [[group isomorphism|isomorphism]] between {{math|''G''}} and {{math|''H''}} is a [[bijection|bijective]] {{gli|homomorphism}} from {{math|''G''}} to {{math|''H''}}, that is, a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. Two groups are ''isomorphic'' if there exists a group isomorphism mapping from one to the other. Isomorphic groups can be thought of as essentially the same, only with different labels on the individual elements.}}
{{glossary end}}