Timeline for Minimum number of generators of a finite group
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Dec 12, 2019 at 2:02 | answer | added | Will Asness | timeline score: 2 | |
| Oct 23, 2016 at 12:54 | comment | added | Geoff Robinson | I think you could state it this way, not that it differs much from what is said already. Let $d(X)$ denote the minimum number of generators of a finite group $X$. Then $d(G) = 1 + d(G/N)$ if and only if some generating set of cardinality $d(G)$ of $G$ contains an element of $N.$ Otherwise $d(G) = d(G/N).$ | |
| Oct 21, 2016 at 13:40 | comment | added | Derek Holt | The answer to your question is that they differ by $0$ or $1$, and both are possible. You could say more under certain conditions. For example, if $G/N$ is not cyclic and $|N|$ is divisible by at most two primes, then the generator numbers are equal. But you need to ask a more precise question if you want a more detailed answer. | |
| Oct 21, 2016 at 11:43 | comment | added | Mojtaba Jazaeri | what we can say about the exact minimum number of generators? | |
| Oct 21, 2016 at 11:40 | comment | added | Wojowu | I believe Schur-Zassenhaus implies the number of generators of $G$ is at most the sum of the number of generators of $N$ + the number of generators of $G/N$. On the other hand, any set of generators of $G$ induces a set of generators of $G/N$. So the two numbers differ by at most $1$. | |
| Oct 21, 2016 at 11:32 | history | asked | Mojtaba Jazaeri | CC BY-SA 3.0 |