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Any subgroup with exactly two cosets is invariant under conjugation
Subgroup of Index 2 is Normal: Let be a group and let be a subgroup. If the index of in is , then is normal in .
Equivalently: if there are exactly two left cosets of in , then the left cosets equal the right cosets, so for all .
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
Let be a subgroup of a group . The index of in , denoted , is the cardinality of the set of left cosets of in :
(Equivalently, it is the number of distinct right cosets.)
For finite , Lagrange's theorem implies , so in particular divides . Small index has strong consequences: iff , and if then is normal, i.e. a normal subgroup.
Let be a group and let be a subgroup. The subgroup is normal in (written ) if for every ,
Normality says that is stable under the conjugation action of on itself. Equivalently, is normal iff every left coset of equals the corresponding right coset, and this is exactly the hypothesis needed to form the quotient group .
A left coset of a subgroup in a group is a subset of of the form
for some . A right coset is a subset of the form
Left cosets are the equivalence classes of the equivalence relation on defined by iff (equivalently, ). In particular, the set of left cosets forms a partition of , and the number of distinct left cosets is the index . If is a normal subgroup, then for all , and the set of cosets is the underlying set of the quotient group .