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18h
comment Structure of frobenius groups.
If you are looking for general information besides the wikipedia page, the books Huppert: Endliche Gruppen, Dixon & Mortimer: Permutation groups and D. Robinson: A Course in the Theory of Groups cover them with all the import properties mentioned there. As regarding a classification, in some sense the structure of such groups is quite restrictive and there are many different characterisations. So in some sense they are classified as groups with those properties. But there are many examples among permutation groups, linear groups and so on.
1d
comment If $G$ acts $k$-transitive and $k > 5$ and $G$ is neither alternating nor symmetric, then $(n-k)! \ge 2n$
@DerekHolt If I can choose in $S_{12}$ a subgroup $G$ of index $15$ that is $6$-transitive then it will be a counter-example. The only possiblity that came to my mind is that there does not exists such a $G$ in $S_{12}$, is this true? But I also see no theorem or anything else in the book to exclude that...
1d
comment If $G$ acts $k$-transitive and $k > 5$ and $G$ is neither alternating nor symmetric, then $(n-k)! \ge 2n$
@DerekHolt Ok, but still I cannot draw the conclusion, I must miss something. For example if $n = 12, k = 6$ with $t = |S_n : G|$ all of what you said give that $t$ divides $(n-k-1)! = 120$, $t < 2n = 24$ and $n! \le t!$, and this works for example for $t = 15$ or $t = 20$...
1d
comment If $G$ acts $k$-transitive and $k > 5$ and $G$ is neither alternating nor symmetric, then $(n-k)! \ge 2n$
@DerekHolt Do you really mean $k$ in your last sentence, or $S_n$ embeds into $S_m$ with $m = |S_n : G|$? I do not see this relation for $k$, as I could not restrict $k$ enough to make this work, for example we could have $k < (n-k)!$ or $(n-k)! \le k$...
1d
asked If $G$ acts $k$-transitive and $k > 5$ and $G$ is neither alternating nor symmetric, then $(n-k)! \ge 2n$
1d
comment When is an ideal also a ring, and could then be anything said about its relation to the original ring
Great. Thank you!
2d
comment When is an ideal also a ring, and could then be anything said about its relation to the original ring
Okay, just one last question: $eR$ is not a two-sided ideal in general, just in this case?
2d
comment When is an ideal also a ring, and could then be anything said about its relation to the original ring
Is it $eR = (1-e)Re + eRe = (1-e)Re + eR$, hence $(1-e)Re \le eR$?
2d
comment Why is it that in $\mathbb{F}_q \setminus 0$, there are exactly as many squares as non-squares?
What does $a^{(q-1/2)}$ mean? Of course if not every element is a square writing something like $a^{1/2}$ would not be meaningful?
2d
comment When is an ideal also a ring, and could then be anything said about its relation to the original ring
Sorry, I do not see that $e \in eR$ implies $(1-e)R \subseteq eR$?
2d
comment When is an ideal also a ring, and could then be anything said about its relation to the original ring
Why do we have $(1-e)Re \subseteq eR \cap (1-e)R = 0$; that $(1-e)Re \subseteq (1-e)R$ is clear, but why $(1-e)Re \subseteq eR$? Also is the requirement $eR(1-e) = 0$ essential?
Feb
7
accepted Structure of induced module for subgroup $H \le G$, confused by mixing up notions of $FH$-submodule and $F$-subspace
Feb
7
accepted When is an ideal also a ring, and could then be anything said about its relation to the original ring
Feb
6
revised Structure of induced module for subgroup $H \le G$, confused by mixing up notions of $FH$-submodule and $F$-subspace
edited body
Feb
6
asked Structure of induced module for subgroup $H \le G$, confused by mixing up notions of $FH$-submodule and $F$-subspace
Feb
6
comment $H$ be a proper subgroup of finite group $G$ such that $H \cap gHg^{-1}=\{e\} , \forall g \in G \setminus H$ , then $|\cup gHg^{-1}|>\dfrac 12 |G|+1$
Yes. But your argument is shorter and maybe more in the spirit of thinking about Frobenius groups. As I started learning and reading about them in papers I often encountered arguments (and it still happends...) which I did not understood and later I realised that they depend on such "high level" properties as nilpotency of kernel, uniqueness of Frobenius representation and action up to equivalence, and of course that the kernel is a subgroup. So I guess these are the properties that should come to mind if Frobenius groups are encountered.
Feb
6
comment When is an ideal also a ring, and could then be anything said about its relation to the original ring
Proves how sensitive these notions are to context and definition. But as said my first two paragraphs are just meant as a motivation and do not relate that much to the actual question, which is concerned with ideals. Maybe I should consider deleting the part about subrings...
Feb
6
comment When is an ideal also a ring, and could then be anything said about its relation to the original ring
I added $1 \in S$. So now it should be clear. But anyway, the particular definition of a subring is not that important for my question. As I am asking when ideals are rings with a possible different multiplication and possible different identity.
Feb
6
revised When is an ideal also a ring, and could then be anything said about its relation to the original ring
added 12 characters in body
Feb
6
comment When is an ideal also a ring, and could then be anything said about its relation to the original ring
I understood the definition given on wikipedia that way, and I think it is pretty clear that they mean the same unity in the substructure. Also as written in S. Lang Algebra he defines subring: "A subset $B$ of a ring $A$ is called a subring if it is an additive subgroup, if it contains the multiplicative unit, and if $x,y \in B$ implies $xy \in B$". Maybe there are other definitions around.