Question 4 - A-Level Maths

Prove that, for a group of order 10, every proper subgroup must be cyclic. The set $M = \{ 1,2,3,4,5,6,7,8,9,10 \}$ is a group under the binary operation of multiplication modulo 11.
Show that $M$ is cyclic.
List all the proper subgroups of $M$. The group $P$ of symmetries of a regular pentagon consists of 10 transformations $$\{ \mathrm { A } , \mathrm {~B} , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm {~F} , \mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm {~J} \}$$ and the binary operation is composition of transformations. The composition table for $P$ is given below.
A B C D E F G H I J
A C J G H A B I F E D
B F E H G B A D C J I
C G D I F C J E B A H
D J C B E D G F I H A
E A B C D E F G H I J
F H I D C F E J A B G
G I H E B G D A J C F
H D G J A H I B E F C
I E F A J I H C D G B
J B A F I J C H G D E
One of these transformations is the identity transformation, some are rotations and the rest are reflections.
Identify which transformation is the identity, which are rotations and which are reflections.
State, giving a reason, whether $P$ is isomorphic to $M$.
Find the order of each element of $P$.
List all the proper subgroups of $P$.