Properties of Cyclic Groups Proposition 3 Let G

Properties of Cyclic Groups • Proposition 3: Let (G, ) be a cyclic group with |G| = n. Then (G, ) is isomorphic to (Zn, ). • We can extend the notion of cyclic subgroups to elements not of finite order in the following way. Definition: Let (G, ) be a group, and let a G. Then the set <a> = {an : n Z} forms a subgroup, called the cyclic • • subgroup generated by a. If G = <a> for some element a G, then G is said to be a cyclic group, and a is said to be a generator of G. Proposition 4: Let (G, ) be an infinite cyclic group. Then (G, ) is isomorphic to (Z, +).

Properties of Cyclic Groups - continued Definition and Notation: If (G, ) is a finite group, and H is a subgroup of G, we know that |H| |G|, i. , e. , |G|/|H| is a positive integer. This number is called the index of H in G, notation [G: H]. Thus, a consequence of Lagrange’s Theorem is that |G| = [G: H]|H| if H is a subgroup of the finite group G. • Proposition 5: a) Every subgroup of a cyclic group is cyclic. b) Let G be a finite cyclic group with |G| = n, and let m be a positive integer such that m n. Then G has a unique subgroup H with |H| = m. c) A finite group (G, ) with |G| > 1 has no proper subgroups if and only if |G| = p, a prime. •

Applications of Lagrange’s Theorem • Theorem 2 (Fermat’s Little Theorem): Let p be a prime and let a be an integer. Then ap a (mod p). • Theorem 3 (Euler’s Theorem) : Let n be a positive integer and let a be an integer relatively prime to n. Then a (n) 1 (mod n).