Answer to Question #128662 in Abstract Algebra for Jai

Question

Show that if g is a non cyclic group of order n then g has no elements of order n. Further give an example with justification of a non cyclic group all of whose proper subgroups are cyclic

Answer 1

$$Solution$$

If $G$ contains element $g$ of order $n$ , then $G$ contains cyclic subgroup $<g>$ of order $n$ .

Then $∣G∖<g>∣=∣G∣−∣<g>∣=n−n=0 (G∖<g>$ is complement $<g>$ in $G$ ), that is $G=<g>$ is a cyclic group. So if $G$ is a non-cyclic group of order $n$ , then $G$ has no element of order $n$ .

$Z_2​⊕Z_2$ is a non-cyclic group with cyclic proper subgroups

{$(0,0),(0,0),(0,1),(0,0),(1,0),(0,0),(1,1)$ }

$Z_2​⊕Z_2​$ does not have other proper subgroups, because

$⟨(1,0),(0,1)⟩=⟨(1,0),(1,1)⟩=⟨(1,1),(0,1)⟩=Z_2⊕Z_2​$

So every proper subgroup of $Z_2⊕Z_2​$ ​ is a cyclic group.