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 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 $\frac{(n-1)!}{|Aut(G)|}$. Mar 10 comment Give an example where $A \subseteq B$ with $A \neq B,$ but $\left\langle A\right\rangle= \left\langle B\right\rangle.$ @Hayden: Why non-trivial? ;-) Mar 6 comment Enumerating double coset representatives in the symmetric group on a vector space @DerekHolt: Thank you for this info. I should have thought of O'Nan-Scott myself. Mar 5 revised Enumerating double coset representatives in the symmetric group on a vector space moved text to improve what's seen in listings Mar 5 asked Enumerating double coset representatives in the symmetric group on a vector space Mar 5 comment Suppose $G$ is a group and $a\in G$ with $|a| = m$. Prove that $\langle a^k \rangle=\langle a \rangle \iff \gcd(k, m) = 1$. I don't see a difference in the level of the answers for this problem, only in style. But anyway, let's agree to disagree. I'll keep liking (and upvoting) your answers to difficult problems, and disliking (but not downvoting) your answers to easy problems. Mar 5 comment Suppose $G$ is a group and $a\in G$ with $|a| = m$. Prove that $\langle a^k \rangle=\langle a \rangle \iff \gcd(k, m) = 1$. I like your posts about intermediate to difficult problems where an experienced mathematician (like you are) has to think a bit about a solution. There shortness is a plus. But a beginner knowing how to manipulate equivalences can prove just easy things. I think learning the proper techniques that work also for difficult problems is worth more than being able to solve a trivial problem more quickly. (Universal properties are a good thing to know about, but many - not you - on mse don't have the technique to apply them.) Mar 5 comment Suppose $G$ is a group and $a\in G$ with $|a| = m$. Prove that $\langle a^k \rangle=\langle a \rangle \iff \gcd(k, m) = 1$. +1 for the good style Mar 5 comment Suppose $G$ is a group and $a\in G$ with $|a| = m$. Prove that $\langle a^k \rangle=\langle a \rangle \iff \gcd(k, m) = 1$. You proof is nicely short, but I wouldn't teach math beginners this proof style. Matt's style is the way to go (it generalizes also to harder problem) at least for beginners' questions. For advanced questions style wouldn't matter so much, but equivalences might not work out anymore. Mar 5 awarded Organizer Mar 5 comment Subgroup structure of $\mathrm{SL}(2, p^2)$ and its irreducible characters A semi-obvious subgroup is the quaternion group. Less obvious is $SL_2(3)$, see math.stackexchange.com/questions/1054917/…. Mar 5 revised Is $SL(2, 3)$ a subgroup of $SL(2, p)$ for $p>3$? added tag finite-groups Mar 5 suggested approved edit on Is $SL(2, 3)$ a subgroup of $SL(2, p)$ for $p>3$? Mar 5 answered Is $SL(2, 3)$ a subgroup of $SL(2, p)$ for $p>3$? Mar 2 comment Group presentation: How can we determine the group with the presentation below? Forget my last comment. Just try to find a semi-direct product of two cyclic groups of order $4$ that is the image of your group $G$. This shows $|G|\ge 16$, and $|G|\le 16$ you know by my first comment. Hence $G$ is isomorphic to that semi-direct product.