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 Yearling
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Mar
15
comment computing the orbits for a group action
Do you know the number of fixed points and the order of your group?
Mar
13
comment Computing the order of a group element
My first comment shows that my other comments are wrong: To factor $p=2qr+1$ just break the RSA with modulus $qr$. The order of elements in $\mathbb{Z}/qr\mathbb{Z}^\times$ give divisors of $\lambda(qr)$ that allow factoring $qr$.
Mar
13
comment Let $H,K$ be finite groups , if for any finite group $G$ , $h(G,K)=h(G,H)$ holds , then is it true that $i(G,H)=i(G,K)$ for any finite group $G$ ?
@SaunDev: So you can rephrase your question slightly simpler in the style of math.stackexchange.com/questions/1161733/…
Mar
12
comment $G$ is a simple group of order $60$.Then $G$ contains a subgroup of order 12
$n_2 = 3$ gives you a non-trivial homomorphism to $S_3$.
Mar
12
comment Can I recover a group by its homomorphisms?
@Turion: Five minutes wikipedia (en.wikipedia.org/wiki/Residually_finite_group) and you'll know enough about it (e.g., that you should have included this condition from the beginning in your question). It is as essential as "finitely generated": f.g. prevents that your numbers are infinite and r.f. prevents that they are all zero.
Mar
12
comment Computing the order of a group element
@DRF: We seem to agree. My comment with $p=2qr+1$ was about the fact, that you won't get lucky often enough.
Mar
12
comment Computing the order of a group element
@DerekHolt: What use does an order-finding algorithm have if you can get only elements of order $p-1$ or $\frac{p-1}{2}$?
Mar
12
comment Computing the order of a group element
I somehow doubt that your memory is correct: If $p = 2qr+1$ where $q$ and $r$ are primes both of about the same size, then one is unlikely to find elements of order $q$ without knowing $r$. Maybe you read the implication the other way round (which is true)?
Mar
12
comment If G is a group such that all of its proper subgroups are abelian, then G itself must be abelian
$Q_8\times\mathbb 1\times 1\le Q_8\times\mathbb Z_2\times\mathbb Z(3^{\infty})$ is abelian?
Mar
12
comment Computing the order of a group element
If you take $Z_n^\times$ for $n$ a product of two primes instead of $n$ prime, then being able to find the order of group elements quickly implies that you can break the RSA for modulus $n$.
Mar
12
comment Let $H,K$ be finite groups , if for any finite group $G$ , $h(G,K)=h(G,H)$ holds , then is it true that $i(G,H)=i(G,K)$ for any finite group $G$ ?
Rereading your question I wondered if you considered the implications of $i(G,H) = i(G,K)$ for $G$ being $H$ or $K$?
Mar
12
comment Can I recover a group by its homomorphisms?
@Turion: I think it is your question, so you deserve the credit for it being a good question.
Mar
12
comment Can I recover a group by its homomorphisms?
@Turion: You could post it as another question (maybe without the conjugation). I'd be interested in knowing the solution.
Mar
12
revised Can I recover a group by its homomorphisms?
missed a wrong $n_G$ in last edit
Mar
12
comment Let $H,K$ be finite groups , if for any finite group $G$ , $h(G,K)=h(G,H)$ holds , then is it true that $i(G,H)=i(G,K)$ for any finite group $G$ ?
$i(G,H) = h(G,H)-\sum_{N \lhd G, N\ne 1 } i (G/N , H)$, so induct on the order of $G/N$.
Mar
12
awarded  Yearling
Mar
11
comment Show that under certain conditions the factors of direct product are isomorphic
@Stefan: You could look also at the projection from $G$ to the first coordinate, and restrict it to $D$.
Mar
11
comment Can I recover a group by its homomorphisms?
@Turion: Do you have a clue about the answer to your question for finitely generated residually finite groups?
Mar
11
revised Can I recover a group by its homomorphisms?
corrected $n_G$ to $n_H$
Mar
10
comment How many conjugates does a regular permutation group have?
Using my last comment one should be able toarrive at the conclusion that the desired number is $\frac{(n-1)!}{|Aut(G)|}$.