Answer to Question #270889 in Abstract Algebra for kd bhai

Question

Show that the set A = { 1,2,4,5,7,8 }with an operation as multiplication modulo 9 is a cyclic group.

Find the order of various elements and subgroup generated by them.

Answer 1

If there exists a group element g ∈ G such that $\langle$ g$\rangle$ = G, we call the group G a cyclic group. We call the element that generates the whole group a generator of G.

we have:

$\langle2\rangle=\{2,4,8,7,5,1\}=A$

so A is  cyclic group

$|2|=6$

$\langle4\rangle=\{4,7,1\}$

$|4|=3$

$\langle1\rangle=\{1\}$

$|1|=1$

$\langle5\rangle=\{5,7,8,4,2,1\}$

$|5|=6$

$\langle7\rangle=\{7,4,1\}$

$|7|=3$

$\langle8\rangle=\{8,1\}$

$|8|=2$