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## #1 2010-04-29 10:22:53

GroupTheorist
Guest

### What are the orders of the elements in (Z_11 - {0}, x)

Hey,

I'm a little confused on this question: What are the orders of the elements in (Z_11 - {0}, x)?

From the previous example, it said

The element 1 has order 1
As 2=2,  2²=4, 2³=1, the order of 2 is 3

And so on, and I understand that we are meant to find when a^m=e (where e is the identity of the group, and it makes sense that the identity of this multiplicative group is 1).

However, when I try to apply this to the question above,

I get:

2=2, 2²=4, 2³=8, 2^4=5, 2^5=10, 2^6=9, 2^7=7, 2^8=1

So should 8 be the order of 2? The answer in the book says that it is 10?

Can someone explain why this is the case, please?

Thank you very much in advance.

## #2 2010-04-29 13:15:20

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: What are the orders of the elements in (Z_11 - {0}, x)

It looks like you may be computing the powers 2^n, and then reducing them modulo 11.  While this works, it is inefficient.  You can use the reduced numbers instead:

2*2 = 4
2*4 = 8
2*8 = 16 = 5
2*5 = 10
2*10 = 20 = 9
2*9 = 18 = 7
2*7 = 14 = ?

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #3 2010-04-30 02:19:17

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

### Re: What are the orders of the elements in (Z_11 - {0}, x)

GroupTheorist wrote:

2=2, 2²=4, 2³=8, 2^4=5, 2^5=10, 2^6=9, 2^7=7, 2^8=1

So should 8 be the order of 2? The answer in the book says that it is 10?

Are you familiar with Lagranges theorem? One consequence of Lagrange is that the order of an element of a finite group must divide the order of group. The group youre dealing with is of order 10; hence the possible orders for 2 are 1, 2, 5 and 10. You therefore only need to check 2[sup]1[/sup], 2[sup]2[/sup], 2[sup]5[/sup] and 2[sup]10[/sup].

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## #4 2010-04-30 02:52:10

abbie stockley
Member
Registered: 2010-04-30
Posts: 1