# Find the number of generators of a cyclic group with a given order n [duplicate]

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Let us say that $$n = 5$$. The number of generators of a finite cyclic group would be the number of numbers that are relatively prime to $$n$$ and the identity element.

Here, when $$n=5$$, the number of generators would be $$2,3,5$$. We would also have to include the identity element generator, $$e$$ for each finite cyclic group. Thus the number of generators of a cyclic group with order $$5$$ would be 4.

Similarly, when $$n=12$$, the number of relatively prime generators would be $$2,3,5,7,11$$ and $$1$$ for a total of 6.

• I think I understand how to find the number of generators of a cyclic finite group, but I can't really explain why.

• I also can't draw the connection to this statement:

The order of an element in a finite cyclic group is the smallest positive integer $$n$$ such that $$a^n=e$$, denoted ord $$a$$.

• How would I prove that the order of an element $$a \in G$$ equals the order of $$G$$? What am I failing to consider?

## marked as duplicate by Dietrich Burde abstract-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Apr 9 at 16:27

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• Do you mean, if $G=\langle a \rangle$, then to prove $|a|=|G|$ ? – Chinnapparaj R Apr 9 at 15:51
• You are confusing a number of different things and you might need to improve your notation to make yourself clear. The identity element in a group is only a generator of the whole group if the group itself has order $1$ - ie it is the trivial group. You need to tell us what the numbers $2,3,5$ mean - you say these are "the number of generators" - that doesn't make much sense as you've written it - are these elements of the group, or something else? – Mark Bennet Apr 9 at 15:52
• @MarkBennet The question says "Find the number of generators of a cyclic group having the given order 8". I would answer with "The number of generators of a cyclic group with order 8 is 4". I am sorry I can't explain much further, that is why I originally posted, to try to get better organized thoughts and lay out my thinking. – Evan Kim Apr 9 at 19:50

## 1 Answer

A cyclic group of order $$n$$ has $$\phi(n)$$ different generators, see these duplicates:

Cyclic Group Generators of Order $n$

How many generator has a cyclic group of order n?

How to find a generator of a cyclic group?