Timeline for A finite group whose conjugacy classes outside a normal subgroup have equal size
Current License: CC BY-SA 4.0
11 events
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| Mar 31, 2023 at 5:02 | comment | added | David E Speyer | Another example is the group of affine linear transformations of $\mathbb{F}_q$, for some finite field $q$. The translations $x \mapsto x+b$ form a normal subgroup; for any $a \neq 1$, all the maps of the form $x \mapsto ax+b$ are conjugate to each other, forming a conjugacy class of size $q$. | |
| Mar 31, 2023 at 4:57 | answer | added | KhashF | timeline score: 3 | |
| Mar 30, 2023 at 1:43 | comment | added | Gerry Myerson | Here's an example that isn't covered (I think) by anything posted here yet. The special linear group of $2\times2$ matrices over the field of three elements is a group of order $24$ with a (nonabelian) normal subgroup $N$ isomorphic to the quaternion group of order eight and index three; there are four conjugacy classes outside $N$, each of size four. See people.maths.bris.ac.uk/~matyd/GroupNames/1/SL(2,3).html | |
| Mar 30, 2023 at 1:12 | comment | added | Gerry Myerson | @Joachim, sorry, I was responding to the question and overlooked your comment. | |
| Mar 29, 2023 at 9:12 | comment | added | Yiftach Barnea | Is there analogy for characters? If yes, what is it? | |
| Mar 29, 2023 at 4:04 | comment | added | Joachim König | @GerryMyerson These are all special cases of what I'm mentioning though (not that there aren't any others). | |
| Mar 29, 2023 at 2:37 | comment | added | Gerry Myerson | The alternating group on four letters has a normal subgroup of index three, and the two conjugacy classes outside this subgroup both have size four. The nonabelian group of order $21$ has a normal subgroup of index three, and the two conjugacy classes outside this subgroup have the same size. | |
| Mar 29, 2023 at 2:19 | comment | added | Gerry Myerson | The group of symmetries of a square has several normal subgroups of index two; for each such normal subgroup $N$, the conjugacy classes outside of $N$ have size two. | |
| Mar 29, 2023 at 0:57 | comment | added | Joachim König | Any case with $N$ abelian of prime index is always an example, since the centralizer order of $x$ and of $nx$ ($x\notin N, n \in N$) is the same. | |
| Mar 28, 2023 at 22:27 | answer | added | Yiftach Barnea | timeline score: 2 | |
| Mar 28, 2023 at 21:40 | history | asked | Steve Stahl | CC BY-SA 4.0 |