Exercise
An automorphism of a finite field of order q is also an automorphism of the multiplicative group ( of order q - 1) of that field.
For which prime powers q is it true that the automorphism group A of a finite field F of order q is equal to the automorphism group B of the cyclic group of order q - 1?
For no q at all.
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For all q.
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For q = 2, 4.
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For q = 2, 3, 4.
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No. For q = 2, both A and B are trivial groups.
Yes! It follows readily from equating |A| = a to |B| =
(pa) =
pa - pa-1.
(pa) =
pa - pa-1.
No.
For q = 3, we have A = {e} whereas inversion is a nontrivial element of B.
No.
For q = 3, we have A = {e} whereas inversion is a nontrivial element of B.