# Thread: Non-isomorphic groups of order 42.

1. ## Non-isomorphic groups of order 42.

Is the following argument correct, thanks

I want to find 3 non-isomorphic groups of order 42.

1) C_42 th cyclic group of order 42 is cyclic hence abelian.
2) D_21 Dihedral group of order 42, non-abelain.
3) C_3 \times D_7

C_42 is non-isomorphic to the others, since it is
abelian and the others are non-abelain.

Remains to show that D_21 and C_3 \times D_7 are non-isomorphic.

D_21 consists of 21 reflections of order 2 and 21 rotations of the form D^n.
The possible orders for the rotations are divisors of 21, i.e, 1,3,7, or 21.
So this group has no element of order 6.
Now consider C_3 \times D_7 , C_3 has a generator c which has order 3, and D_7 has a reflection S of order 2, so in C_3 \times D_7 there is an element (c,S) of order lcm(3,2)=6, hence the two groups are non-isomorphic.

2. Originally Posted by peteryellow
Is the following argument correct, thanks

I want to find 3 non-isomorphic groups of order 42.

1) C_42 th cyclic group of order 42 is cyclic hence abelian.
2) D_21 Dihedral group of order 42, non-abelain.
3) C_3 \times D_7

C_42 is non-isomorphic to the others, since it is
abelian and the others are non-abelain.

Remains to show that D_21 and C_3 \times D_7 are non-isomorphic.

D_21 consists of 21 reflections of order 2 and 21 rotations of the form D^n.
The possible orders for the rotations are divisors of 21, i.e, 1,3,7, or 21.
So this group has no element of order 6.
Now consider C_3 \times D_7 , C_3 has a generator c which has order 3, and D_7 has a reflection S of order 2, so in C_3 \times D_7 there is an element (c,S) of order lcm(3,2)=6, hence the two groups are non-isomorphic.
It looks good to me. For these sort of problems there are many different ways to solve them.