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GroupTheorist

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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.

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 = ?

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"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..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

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

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].

abbie stockley

Member

Registered: 2010-04-30

Posts: 1

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

hi what is your name