Order of Group G Generated by a,b Relation

[ElDavidas]

The question reads :

"What is the order of a group G generated by elements a and b subject only to the relations

$$a^7 = 1$$ , $$b^3 = 1$$ , $$ba = a^rb$$"

I know that the order is the number of elements in the group.

I'm having a lot of trouble answering a lot of these questions.
Any help would be greatly appreciated.

Thanks in advance.

 

[Hurkyl]

Well, you could always write down all of the elements of the group.

 

[ElDavidas]

Well, you could always write down all of the elements of the group.

How do you go about doing that using the information given? I don't know where to begin.

Also, does $a^7 = 1$ mean a to the 7 is equal to the identity element?

 

[Hurkyl]

Well, there's a, and aa, and aaa, and ab, and ba, and aba, and bab, and abbaabbab, and...


Yes, the relation a^7 = 1 means that aaaaaaa is the identity.

 

[ElDavidas]

I think I see where this is going.

Does $a^8 = a^7 * a = 1 * a$ make sense?

 

[topsquark]

I think I see where this is going.

Does $a^8 = a^7 * a = 1 * a$ make sense?

Yup. And use the ba relation to reorder your products so that they all read "aaaa...bb..."

-Dan