Study Notes on Cyclic Groups and Their Properties

Cyclic Groups

Definition of Cyclic Groups

• Let G be a group.
• A subgroup of G is a cyclic subgroup if it can be generated by a single element a.
• Notationally, is defined as {a^n : n ∈ ℤ}.

Properties of Cyclic Groups

• If G is cyclic, it can be expressed as:
  1. G = (where a is a generator).
  2. a is a generator for G.
  3. G is abelian (commutative).

Examples of Cyclic Groups

Example 1: Group of Integers

• Define U = { Z | Z = 1,3 }, using additive notation.
• When considering the additive group of Z, every element can be expressed as an integer multiple of 1.
• Every element n ∈ Z can be expressed as n = k × 1 where k ∈ ℤ.
  • Notation: = {na : n ∈ ℤ} implies that Z is cyclic.
• The integers 1 and -1 are both generators of Z.
• There are no other distinct generators for the group Z.

Example 2: Finite Cyclic Groups

• Consider Z_n = {0, 1, 2, …, n-1} under addition modulo n:
• Example when n = 4:
  • (trivial subgroup)
• The entire group Z_4 is cyclic since it can be generated by 1 or 3.

Non-Cyclic Groups

• Example: Consider the Klein four-group V₄ = {e, a, b, c}
  • The operation tables yield:
  • e multiplied by any group element returns that element.
  • Each element combines with itself to return e, indicating no generators sufficient to form a single cycle.
• Since not all elements can generate the entire group, V₄ is not cyclic.

The Division Algorithm for Two Integers

Theorem: Division Algorithm

• For any integers n and m (with m > 0), there exist unique integers q and r such that:
  • $n = mq + r$
  • where $0 \le r < m$.
• Example 1:
  • Let n = -13 and m = 8:
  • Applying the division algorithm, we have:
    • $-13 = 8 imes (-2) + 3$

Properties of Cyclic Groups (Continued)

Every Cyclic Group is Abelian

• Let G be a cyclic group generated by .
• For any two elements x = a^m and y = a^n in G:
  1. The operation between any two elements yields:
  • $xy = a^m a^n = a^{m+n}$
  • which shows that the order of multiplication does not affect the result, hence G is abelian.

Subgroups of Cyclic Groups

• If H is any subgroup of a cyclic group G, then H is cyclic as well.
• If H = {e} (the trivial subgroup), it is cyclic.
• If H is non-trivial,
  • Each element in H can be expressed as powers of a specific generator of G.
  • Let be the cyclic group, and if H is non-trivial, then it contains elements of the form a^k (for some integer k).

Fact about Subgroups of Z

• All subgroups of (the group of integers under addition) are cyclic.
• Any subgroup can be represented in the form H = nZ where n ∈ ℤ.