/sci/ - Science & Math » Thread #7626397

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Anonymous Fri 30 Oct 2015 15:03:21 No.7626397 View Reply Report Delete

Quoted By: >>7626544 >>7626574 >>7626609

Can someone help me understand how the group $\mathbb{Z}/n\mathbb{Z}$ is the same thing as the cyclic group of n elements?

>pic irrelevant

Anonymous

Anonymous Fri 30 Oct 2015 15:07:46 No.7626412 Report Delete

Quoted By:

Write down the injection and find out if it's a homomorphism and invertible.

Anonymous

Anonymous Fri 30 Oct 2015 16:32:32 No.7626544 Report Delete

Quoted By:

>>7626397
1 is the generator

Anonymous

Anonymous Fri 30 Oct 2015 16:46:42 No.7626574 Report Delete

Quoted By: >>7626609

>>7626397
Because it's cyclic (0 is an element, 1 is an element, 2 is, ..., n - 1 is an element, n is not because it falls into 0)
Because it has n elements.

My discrete math class only went over the congruence classes of the integers modulo n so this may not help. Hope it does though.

Anonymous

Anonymous Fri 30 Oct 2015 17:00:34 No.7626609 Report Delete

Quoted By: >>7626621

>>7626574

CS majors don't belong here, leave.

>>7626397

Because a^m is short hand for a∘a∘...∘a (m times) where ∘ is the group operation. When ∘ is addition or some other abelian operation then a^m is written as ma or m∗a.

Anonymous

Anonymous Fri 30 Oct 2015 17:05:53 No.7626621 Report Delete

Quoted By:

>>7626609
Math major here, actually. We went over fields and rings more than we went over groups, shame on us.

Anonymous

Anonymous Fri 30 Oct 2015 18:13:02 No.7626747 Report Delete

Quoted By:

You can prove that any cyclic group of order n is isomorphic to Z/nZ and any infinite cyclic group is isomorphic to Z. Can you think of what the isomorphism should be?

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