- #1

sa1988

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**EDIT: I've just realized this is the 'Calculus and beyond' subforum - I saw 'beyond' and thought, "Well I've done all my calculus, and now I'm doing group theory, so this thread must go here!". But now I realize it surely belongs somewhere else. Sorry about that. Mods feel free to shift it to the right place...**

I'm new to group theory and am sure this is a very simple question as I'm only on Week 1 of my Group Theory module at university. I may even have got the correct answer (apart from one bit I'm stuck on), but I'd like to check it all before next week's lectures come around, just to be sure I'm on the right track with the various definitions and concepts.

1. Homework Statement

I'm new to group theory and am sure this is a very simple question as I'm only on Week 1 of my Group Theory module at university. I may even have got the correct answer (apart from one bit I'm stuck on), but I'd like to check it all before next week's lectures come around, just to be sure I'm on the right track with the various definitions and concepts.

1. Homework Statement

Write out the multiplication table for the group G = ℤ

_{7}

^{*}

List all the inverses of the elements of G.

Find an element of G such that every member of G is a power of that element

## Homework Equations

## The Attempt at a Solution

So we were told that a group ℤ

_{n}

^{*}is defined such that it contains all integers except 0, and that all values in the group can only exist if they have no common factor with n. (e.g. ℤ

_{4}

^{*}= {1,3} . 2 is omitted because it is a factor of 4)

Hence the group G = ℤ

_{7}

^{*}= G{1,2,3,4,5,6,7}

Also, I understand a numerical group such as this uses the modulo form of multiplication, whereby any multiple resulting in something greater than n is simply treated as a the remainder. Hard to explain but I'm sure you all know what I mean. e.g. 5*3 with modulo 6 = 3

- So... The Multiplication table for ℤ
_{7}^{*}:

__0 1 2 3 4 5 6__(ignore the 0 - I needed to add it to keep the table aligned)

__1__1 2 3 4 5 6

__2__2 4 6 1 3 5

__3__3 6 2 5 1 4

__4__4 1 5 2 6 3

__5__5 3 1 6 4 2

__6__6 5 4 3 2 1

I'm hoping this is correct.

- Next, the inverse of each element of G:

2→4

3→5

4→2

5→3

6→1

Again I'm hoping this is correct.

- And the final part: "Find an element of G such that every member of G is a power of that element"

Well, this is the bit I'm really unsure about. What does it mean by "every

*member*of G"..? I assumed it was just another word for element, but still I can't see a clear, specific answer. I was thinking the answer is 1, because 1 to any power will give another member of G (assuming a member is also an

*element*)..? I just don't know to be honest. Hopefully it's something blindingly simple.

Thanks for any help, particularly regarding that final part at the end.