Sylow theorems

For finite groups, Sylow theorems constrain existence, conjugacy and number of maximal prime-power subgroups.

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Binary operations
A binary operation combines two elements into one in a specified set.
Normal subgroups
A subgroup invariant under conjugation allows consistent multiplication of cosets.
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.
Field extensions
A field extension contains a smaller field and can be viewed as its vector space.