Find Subgroup of Order 4 in $\mathbb{Z}$ /13

[ElDavidas]

The question reads as:

"Let $G = (\mathbb{Z} /13)^*.$ Find a subgroup H of G such that |H| = 4. "

I think this means that you have to find a subgroup that has order 4. Although I'm not entirely sure what that means in this context.

Any help will be appreciated.

 

[AKG]

Find a subgroup of G with four elements.

 

[ElDavidas]

Find a subgroup of G with four elements.

Ok, I understand this.

I know that for a set to become a subgroup it has to satisfy the operation, contain the identity element and also have an inverse.

How do you go about writing what the possible subgroups are for (Z/13) with order equal to 4? (Which I think is all the congruence classes with mod 13)

I can understand the theory but don't really know how to apply it to a concrete example.

 

[matt grime]

I'm not sure you can claim to understand the theory if you cannot do this question.

Firstly you're talking about (Z/13)^* which is the group of units modulo 13, since 13 is a prime that is the numbers 1,2,3,...,11,12 with multiplcation mod 13.

We are not talking about "all the congruence classes with mod 13" if indeed you did understand the theory you'd know that there are no subgroups of Z/13 under addition (presumably you mean addition, as that is the group operation defined on all the congruence classes) with order 4 since 4 does not divide 13.

Secondly, why don't you just experiment? Pick a number, work out its square, cube, and 4th power and see if you get what you want. There are naively at most 12 things here for you to try, though you obviosuly won't pick 1 or 12, will you?