Cyclic Groups - Abstract Algebra I

Flashcards covering key vocabulary and concepts from the chapter on Cyclic Groups in Abstract Algebra.

Cyclic Group

A group G is called cyclic if there exists an element a in G such that G is generated by a, denoted .

Generator

An element a of a cyclic group G that generates the entire group such that G = .

Theorem 1 - Infinite Order

If x has an infinite order in a cyclic group, then xi = xj if and only if i = j.

Theorem 1 - Finite Order

If x has a finite order n in a cyclic group, then x^k = e if and only if n divides k.

Subgroup of Cyclic Group

Every subgroup of a cyclic group is itself cyclic.

Order of a Group

The number of elements in a group, which is finite if the group has a finite number of elements.

Prime Order

A group has prime order if its only subgroups are the trivial subgroup e and the group itself.

Example of Cyclic Group

The group Z_n under addition is a cyclic group generated by 1.

Fundamental Theorem of Finite Abelian Groups

Each finite abelian group can be expressed as a direct product of cyclic groups of prime power order.

Lagrange's Theorem

The order of a subgroup divides the order of the group.