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Aug
17
revised Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$
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Aug
17
comment Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$
First reduce (by induction) to the case where $b = 1$. Then in that case, the number of subgroups of order $p^{a+1}$ that contain $K$ are in bijection with the subgroups of order $p$ in $N_G(K)/K$.
Aug
7
awarded  Enlightened
Aug
7
awarded  Nice Answer
Aug
4
awarded  Pundit
Aug
4
comment Sequence of primes by concatenating digits in a given base.
@DarthGeek: At least in base 10, there is no example of such an infinite sequence. See mathworld.wolfram.com/TruncatablePrime.html
Aug
4
comment Sequence of primes by concatenating digits in a given base.
$333333331 = 17 \cdot 19607843$
Aug
2
comment Is every Banach space isometrically isomorphic to the dual of a normed space?
math.stackexchange.com/questions/242034/…
Jul
30
comment Characterization of nilpotent groups
@JoelMoreira: It might be a good idea to look up the results about Engel groups, perhaps there is something there that would give you a proof or a counterexample for the bounded class question
Jul
30
comment Characterization of nilpotent groups
@DerekHolt: The survey here mentions at page 4 that Golod has constructed a group that is a finitely generated, non-nilpotent Engel group such that every subgroup generated by two elements is finite. Since finite Engel groups are nilpotent, that should give you an example.
Jul
30
revised Groups of order $n^2$ that have no subgroup of order $n$
better title
Jul
30
comment Groups of order $n^2$ that have no subgroup of order $n$
@DerekHolt: Thanks for saving me the trouble!
Jul
30
comment Groups of order $n^2$ that have no subgroup of order $n$
With GAP I went through groups of order $n^2$, where $n \leq 30$ and $n \neq 24$, and the examples in this answer are the only ones among them (one for $n = 30$ and two for $n = 28$). There are $8681$ groups of order $24^2 = 576$, so going through them manually might take a very long time
Jul
30
awarded  Nice Answer
Jul
30
revised Groups of order $n^2$ that have no subgroup of order $n$
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Jul
30
revised Groups of order $n^2$ that have no subgroup of order $n$
edited tags
Jul
30
revised Groups of order $n^2$ that have no subgroup of order $n$
added 298 characters in body
Jul
30
answered Groups of order $n^2$ that have no subgroup of order $n$
Jul
29
answered Characterization of nilpotent groups
Jul
14
awarded  Enlightened