Timeline for How many non-isomorphic extensions with kernel $S^1$ and quotient cyclic of order $p$?
Current License: CC BY-SA 4.0
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| May 4, 2019 at 14:38 | comment | added | mathstudent | Thanks. Yes, now I understand it clearly. I need to determine the possible presentation of such groups. So from your answer, I understand that for a given $k$ if we can determine the presentation of $G_k$ (following your notation), then all others can also be deduced. It will very helpful if you please help me to determine the presentation of such groups. | |
| May 4, 2019 at 9:08 | comment | added | YCor | For the choice of non-degenerate alternating form, denote by $G_{2\ell}$ the corresponding central extension of $S^1$ by $C_p^{2\ell}$. Then the groups are just $G_{2\ell}\times C_p^m$ for all $\ell,m$ such that $2\ell+m=k$. For instance for $k=4$ these are $G_0\times C_p^4$ (the direct product), $G_2\times C_p^2$, and $G_4$. | |
| May 4, 2019 at 8:14 | comment | added | mathstudent | Thank you very much. So if $k=4$, then there are 3 non-isomorphic groups. Is there any general way to determine all these groups(maybe the presentation of such groups). In general, for a given k is there any way to determine what are these $1+[k/2]$ groups (maybe how to determine their presentation)? | |
| May 4, 2019 at 8:14 | vote | accept | mathstudent | ||
| Apr 30, 2019 at 21:10 | history | answered | YCor | CC BY-SA 4.0 |