Cosets and Lagrange's theorem

Cosets partition a finite group, so a subgroup's order divides the group's order.

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Cyclic groups
A cyclic group is generated by repeated powers of a single element.
Group homomorphisms
A homomorphism preserves group multiplication; its kernel measures failure of injectivity.
Binary operations
A binary operation combines two elements into one in a specified set.
Field extensions
A field extension contains a smaller field and can be viewed as its vector space.
Fields
A field is a commutative ring where every nonzero element has a multiplicative inverse.