I have to find the number of elements in Aut(Z720) with order 6. Please suggest how to go about it.(adsbygoogle = window.adsbygoogle || []).push({});

1) Aut(Z720) isomorphic to U(720) (multiplicative group of units).

2 ) I am using the fundamental theorem of abelian group that a finite abelian group is isomorphic to the direct products of cyclic groups Zn.

In this case, 720=16×9×5.

Therefore, Aut(Z720)≅U(720)≅Z2×Z4×Z4×Z6.

Now, the possible orders of elements in Z2:1,2; Z4:1,2,4; Z6:1,2,3,6.

Using the result defining the order of an element in external direct products:

If 6=Order(a,b,c,d)=lcm(Order(a),Order(b),Order(c),Order(d)) then:

Case 1 : If Order(d)=6 then lcm(Order(a),Order(b),Order(c))=1 or 2.

Using the the result for cyclic groups:

for every divisor d of the order of a cyclic group G, there exists ϕ(d) elements in G with order d.

It seems there are 16 elements. I am not sure though.

Is this the correct way and how to proceed further? Please suggest.

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# Automorphism groups

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