Ch. 4: Cyclic Groups

Def: Cyclic Group

Criterion for a^i = a^j

Corollary 1: |a|=|<a>|

Corollary 2: a^k = e implies |a| divides k

Theorem: <a^k> = <a^(gcd(n,k)>

Corollary 1: Oders of Elements in Finite Cyclic Groups

Corollary 2: Criterion for <a^i> = <a^j> and for |a^i| = |a^j|

Corollary 3: Generators of Finite Cyclic Groups

Generators of Z_n

Fundamental Theorem of Cyclic Groups

Corollary Subgroups of Z_n

Def: Phi function

Number of elements of each order in a cyclic group

Corollary: Number of elements of oder d in a finite group