Mikko Korhonen
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 Aug 20 answered Help to understand proof to show that a function is uniformly continous in a certain interval (Spivak) Aug 19 comment Exercise on Sylow's Theorem in Dummit and Foote Here's a different solution: $G/P$ is cyclic, so it has a normal subgroup of order $5$. Thus $G$ has a normal subgroup of order $3^2 \cdot 5$, which has a normal $5$-Sylow. Thus $G$ also has a normal $5$-Sylow. Similarly $G/P$ has a normal subgroup of order $7$ so $G$ has a normal subgroup of order $3^2 \cdot 7$ and thus a normal $7$-Sylow. Since each Sylow subgroup of $G$ is normal and abelian, $G$ is abelian. Aug 19 answered A Question on Commutators Aug 16 awarded Revival Aug 16 answered A question about finite simple groups Aug 15 comment Show that a periodic, finitely generated, nilpotent group has finite order. You could use the fact that finitely generated nilpotent groups are supersolvable. Or if that's too much, use the fact that in a finitely generated group factors of the lower central series are finitely generated. Aug 15 answered How to prove that $\int_0^{2\pi} \sqrt{1+\cos^2{t}}\; dt>2\pi$? Aug 15 answered Prove that the rank of the block matrix is rank $A$ + rank $B$ Aug 14 comment Abelian Groups and Number Theory A solution exists if and only if $a \equiv b \mod{d}$, where $d = \gcd(m,n)$. For example, $x \equiv 1 \mod{2}$, $x \equiv 2 \mod{4}$ has no solution. If a solution exists, it is unique modulo $\operatorname{lcm}(m,n)$. Aug 13 awarded Revival Aug 12 awarded Nice Answer Aug 12 answered Is there a way to construct non-cyclic groups of any order? Aug 8 revised Why are regular $p$-groups called “regular?” deleted 7 characters in body Aug 8 answered Questions about group of automorphisms Aug 8 answered Why are regular $p$-groups called “regular?” Aug 7 revised What is the smallest $n$ for which the usual “counting sizes of conjugacy classes” proof of simplicity fails for $A_n$? added 16 characters in body Aug 7 asked What is the smallest $n$ for which the usual “counting sizes of conjugacy classes” proof of simplicity fails for $A_n$? Aug 7 comment Extracting Information from Lists in GAP Also, if you want a list of elements with the desired properties, you can use "Filtered(L, c->IsAbelian(c) and Exponent(c)=n and Order(c)=m);". Aug 5 comment How many $p$-elements does $G$ have? @Adeleh: See this question. Specifically, Miller proves in pg 79-80 of "Theory and Applications of Finite Groups" (1916) that $a_p(G) > p^{\alpha+1}$ when $n_p(G) > p+1$. Now we can apply the fact that $a_p(G) = p^{\alpha}(t(p-1) + 1)$ to improve this to $a_p(G) \geq p^{\alpha}(2p-1)$. This improvement is from Miller's paper "Some Deductions From Frobenius's Theorem" (1942), it is near the end of the paper: link to paper. Aug 5 awarded Revival