The Size of an Automorphism Group

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This Demonstration expresses a multiplicative unit group modulo an integer in terms of isomorphic additive groups and then uses the additive groups to compute the total number of automorphisms of the unit group.

Contributed by: Dominic Milioto (June 2015)
Open content licensed under CC BY-NC-SA


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In this Demonstration, represents the multiplicative unit group of integers modulo , and represents the additive group of integers mod .

If , then . Each is isomorphic to an additive group according to the following rules: , , and for ; and for odd prime , . These isomorphisms can be derived by using the Euler totient function, , which gives the number of positive integers less than or equal to that are relatively prime to . For example, consider for odd . We have and therefore , since both groups are cyclic and of the same order. Once the group has been factored into additive groups, the groups are combined according to prime powers. Each prime power set is a Sylow -group, , and these allow the easy determination of according to the formula:

with and .

Any Abelian group such as is isomorphic to a product of Sylow -groups, and the number of automorphisms of is a product of the number of automorphisms of its Sylow -groups.

In the case of , we have . For , the exponents are , , and , with , , , and , , .

This gives .

A similar calculation gives .

Therefore .

By considering the generators needed to generate the Cartesian product , it can be shown that the minimum number of generators needed to generate is simply the number of factors of the Sylow 2-group. The last line of the result reports this number.

For more details and a derivation of the formula above, see [1].

Reference

[1] C. J. Hillard and D. L. Rhea. "Automorphisms of Finite Abelian Groups". www.msri.org/people/members/chillar/files/autabeliangrps.pdf.



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