Reputation
652
Top tag
Next privilege 1,000 Rep.
Create new tags
Badges
9 15
Newest
 Yearling
Impact
~5k people reached

Mar
2
comment Group presentation: How can we determine the group with the presentation below?
Next try to show that both $g^2$ and $h^2$ are in the center of $G$. (I'm not sure yet which group we will get, I just work comment to comment, I hope you don't mind. It might be that the given relations imply additional relations, but our group has at least $8$ elements.)
Mar
2
comment Can I recover a group by its homomorphisms?
@Turion: Yes, thanks, this should be $n_G$ (also in the line below the formula). Induction is on the order of the group $H$.
Mar
2
comment Group presentation: How can we determine the group with the presentation below?
As a first step you could try to prove that all elements of $G$ can be written as $g^ih^j$ with $0\le i\le 3$ and $0\le j\le 3$.
Mar
1
answered Can I recover a group by its homomorphisms?
Feb
28
comment How many conjugates does a regular permutation group have?
The number of conjugates of a subgroup $G$ of $H$ is the index of its normalizer $N_H(G)$ in $H$. Now look at this question.
Feb
27
comment Can I recover a group by its homomorphisms?
@Turion: I hope to have some time for it at the weekend.
Feb
27
comment Structure of the semidirect product decomposition
One has always to worry a bit about the order of both elements (there are different conventions), but otherwise it is as easy as you think.
Feb
27
comment Structure of the semidirect product decomposition
ad 1) Try multiplication (if $G = N\rtimes H$ then you can write each element $g\in G$ uniquely as product $g = nh$ with $n\in N$ and $h\in H$.) ad 2) look at 1) again. Does this give you an idea?
Feb
25
comment Can I recover a group by its homomorphisms?
You get that number by induction on the subgroups of $H$.
Feb
25
comment Can I recover a group by its homomorphisms?
@anomaly: Modulo conjugation by $H$ the number of surjective homomorphism $G\to H$ should be number of possible kernels times the order of $\mathrm{Out}(H) = \mathrm{Aut}(H)/\mathrm{Inn}(H)$.
Feb
25
comment Can I recover a group by its homomorphisms?
@anomaly: What's $K$? My $U$ or my $H$? Anyway, I hope to solve this by looking at $N_H(K)$.
Feb
25
comment Can I recover a group by its homomorphisms?
@AlexWertheim: For $G$ finite the answer should be yes. If you can determine for every finite group $H$ whether it is a quotient of $G$, then the biggest quotient will be $G$ itself. I think you can get the number of surjective homomorphisms $G \to H$ modulo conjugation by induction: the number of all homomorphisms is given and every non-surjective homomorphism has a proper subgroup $U$ of $H$ as image, so you know their number by induction (you'll have to correct this number considering $N_H(U)$ and the conjugates $U^h$).
Feb
24
revised A group whose automorphism group is cyclic
added warning that answer is wrong
Feb
24
suggested approved edit on A group whose automorphism group is cyclic
Feb
24
comment A group whose automorphism group is cyclic
In case you didn't see it: the accepted answer is wrong! Can you maybe un-accept it? Thanks!
Feb
23
comment A group whose automorphism group is cyclic
I thought about it, but couldn't come up with alternatives (I tried variations of Jack Smith's example for an older question by W4cco). I tend to believe that the answer is no, and that the question could be asked at mathoverflow.net (but infinite abelian groups ain't my strength).
Feb
23
comment What's so special about the group axioms?
@Shaun: The distinction between groups and inverse semigroups depending on whether they are distance-preserving or not in your first comment lacks a bit of context (I guess one finds it in the book you linked to).
Feb
23
accepted How many non-isomorphic central extensions of a cyclic group of order $2$ by the Lamplighter group exist?
Feb
23
comment A group whose automorphism group is cyclic
A hint in case you don't find any other automorphism of $(\mathbb{R}/\mathbb{Z})^2$: Look for vector space automorphisms of $\mathbb{R}$ seen as vector space over $\mathbb{Q}$ fixing $\mathbb{Q}\le\mathbb{R}$.
Feb
23
comment What's so special about the group axioms?
You are not the only one not recognizing the importance of these axioms at once. Historically seen it took more than a century before they were generally considered important in mathematics. Shortly ago I attended a lecture by Bernd Fischer (discoverer of three sporadic finite simple groups) who spoke about his first years in university as student. Several objects (like the sympletic group) were introduced to him without even mentioning that they were groups.