Group theory-cyclic grp

Question

Define a group (A, *) as follows:
Let A = {0, 1, 2, 3, ....., 23}
Given, (a * b) = (a + b) mod 24
The number of proper subgroups of A will be equal

The group is cyclic.So there will be a subgroup for each of the divisors of 24 with the same length as the divisor.

So there is a subgroup with lenght 1,2,3,4,6,8,12,24.But since proper is asked 24 is discarded.So ans is 7.

But in key given as 6.

Comments

Actually subgroup with Identity element e and  G is called as Trivial subgroup.

We have here total subgroup = 8

but two ( e and G) will be trivial subgroup

so ,  Total  no of proper subgroup = 8-2 = 6

Answer = 6

Answer 1

Actually, Proper sub groups does not include trivial subgroup. So, they omit 1 also.

Comments

Can u give me a link of formal defnition.

In this site it says oly proper subsets.This includes a set with 1 element also.

http://mathworld.wolfram.com/ProperSubgroup.html
Thankyou

---

In wikipedia they mentioned as both ways are possible. but dont know what to follow

https://en.wikipedia.org/wiki/Subgroup