Correct Way of Writing Extension of Group

Asked Feb 24 '20 at 00:25

Active Feb 24 '20 at 00:41

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Let $G=C_2\times C_m=\{(g^{t_1},h^{t_2})\mid t_1\in \{0,1\}, t_2\in \{0,1,...,m-1\}\}$.

Is it correct to just say that $G$ is an extension of $C_m$ by $C_2$?

Or it is more accurate to say that $G$ is an extension of $H$ by $C_2$ where $H=\{(1,h^{t_2})\mid t_2\in \{0,1,...,m-1\}\}$.

edited Feb 24 '20 at 00:41

Shaun

asked Feb 24 '20 at 00:25

Andy Kang

Related. – Shaun Feb 24 '20 at 00:42
These are both correct and accurate. See the definition. – freakish Feb 24 '20 at 08:09
The first one is not formally correct - you should replace "is" by "is isomorphic to", but this is a very frequently employed abuse of notation. – Derek Holt Feb 24 '20 at 08:10
@DerekHolt no, that's formally correct. Being an extension means there's a short exact sequence $1\to C_m\to G\to C_2\to 1$. No "is isomorphic to" is needed. – freakish Feb 24 '20 at 08:12

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