Cyclic Groups Study Notes

Cyclic Groups

• Definition: A group G is cyclic if there exists an element a in G such that every element in G can be expressed as a power of a.

• Notation: The cyclic subgroup generated by a is denoted as \<a> = {na : n ∈ ℤ}.

Properties of Cyclic Groups

• Every cyclic group is abelian (commutative).

• The subgroup of a cyclic group is also cyclic.

• For any group generated by element a, all elements can be expressed as integer powers of a.

Example of Cyclic Groups

• Use of additive notation under integers:\n - ℤ is cyclic, with 1 and -1 as generators.

• Examples of \<Zn> (where n is the module):

  • \<0> = {0}, \<1> = {1, 2, 3, …}, \<2> = {2, 4, 6, …} under mod n.

Division Algorithm

• The division algorithm states for given integers n and m, there exists unique integers q (quotient) and r (remainder) such that: n = mq + r, where 0 ≤ r < m.

Notable Theorems

• Theorem (Cyclic Groups): Every subgroup of a cyclic group \<a> is of the form \<a^k> for some integer k.

Group Structure

• Structure follows closure: If a, b are in the group, then ab and a^-1 are also in the group.

• Example of non-cyclic group: Klein Group V4.