Normal subgroups

A subgroup invariant under conjugation allows consistent multiplication of cosets.

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Quotient groups
Cosets of a normal subgroup inherit a well-defined group operation.
Sylow theorems
For finite groups, Sylow theorems constrain existence, conjugacy and number of maximal prime-power subgroups.
Binary operations
A binary operation combines two elements into one in a specified set.
Cosets and Lagrange's theorem
Cosets partition a finite group, so a subgroup's order divides the group's order.
Cyclic groups
A cyclic group is generated by repeated powers of a single element.