Topology of Numbers · Problem 7.3.6

The Class Group for Quadratic Forms — Allen Hatcher

Mathematics — Number Theory · cyclic-group-zn-star

Problem

This problem concerns the question of when the group

\mathbb{Z}_n^*

of congruence classes mod

n

of integers coprime to

n

is cyclic. (a) Show that

\mathbb{Z}_2^*

and

\mathbb{Z}_4^*

are cyclic but

\mathbb{Z}_8^*

is not cyclic and deduce that

\mathbb{Z}_{2^r}^*

is also not cyclic when

r > 3

. (b) Show that if

\mathbb{Z}_n^*

is cyclic then

n = 2

4

p^r

, or

2p^r

for some odd prime

p

. Hint:

\mathbb{Z}_n^*

has even order if

n > 2

. (c) The group

\mathbb{Z}_{p^r}^*

is known to be cyclic when

p

is an odd prime. Show that this implies that

\mathbb{Z}_{2p^r}^*

is cyclic.