Slade Higgins

2022-04-29

Prove that $n\mid \varphi ({a}^{n}-1)$ in "Topics in Algebra 2nd Edition" by I. N. Herstein. Any natural solution that uses $Aut\left(G\right)$

morpheus1ls1

Beginner2022-04-30Added 22 answers

Step 1

Note that$\varphi ({a}^{n}-1)$ measures the number of automorphisms of $\frac{\mathbb{Z}}{(an-1)}\mathbb{Z}$ .

There is a subgroup of order n in this group: if$\varphi$ is the automorphism sending 1 to a, then $\varphi$ generates a subgroup of order n. The statement follows from Lagrange's Theorem.

Note that

There is a subgroup of order n in this group: if

How to find out the mirror image of a point?

Generators of a free group

If G is a free group generated by n elements, is it possible to find an isomorphism of G with a free group generated by n-1 (or any fewer number) of elements?How many 3/4 Are in 1

Convert 10 meters to feet. Round your answer to the nearest tenth

6. Reduce the following matrix to reduced row echelon form:

Let v be a vector over a field F with zero vector 0 and let s,T be a substance of V .then which of the following statements are false

Describe Aut(Zp), the automorphism group of the cyclic group Zp where p is prime. In particular find the order of this group. (Hint: A generator must map to another generator)