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I have a theorem stating that "a semidirect product of a cyclic group of prime order by an abelian group satisfies a certain property". (The property is given in the theorem and according to the terminology of the text where the theorem is stated it is possible to identify that the cyclic subgroup of prime order is the normal subgroup).

Does that mean that the relevant property is satisfies by any semidirect product between two such groups? (i.e. regardless of what type of homomorphisms is defined when defining the semidirect product)

So if I see a semidirect product like $\mathbb{Z}_q \rtimes_{\varphi} \mathbb{Z}_{p^2}$, can I conclude that the result holds regardless of ${\varphi}$, where $p,q$ are distinct primes.

Thanks a lot in advance.

Buddhini Angelika
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  • Yes, you can. If the theorem required a property of the homomorphism, it would (or should) state that explicitly. (Of course without context it is impossible to rule out that the theorem might be wrong.) – Mees de Vries Aug 28 '18 at 09:10

1 Answers1

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The answer is yes. It might just be safer to state "Every semidirect product ..."

J.-E. Pin
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