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Jul
31
answered What is the standard notation to represent the set of primes?
Jul
30
comment A group $G$ with $G'$ abelian and every abelian normal subgroup finite
@MarshalKurosh: In this case $G_p$ itself is an abelian, normal, infinite subgroup of $G_p$. So the assumptions of the statement are not satisfied.
Jul
29
comment A group $G$ with $G'$ abelian and every abelian normal subgroup finite
I don't understand why you need the stuff with $Z(G)$. To me it seems that your proof works when you just let $A$ be any abelian normal subgroup of $G$ containing $G'$.
Jul
28
comment A set which satisfies all conditions for a Group except associativity
A loop is not required to have two-sided inverses.
Jul
28
comment A set which satisfies all conditions for a Group except associativity
Yet $0 - x = x$ if and only if $x = 0$
Jul
28
comment A set which satisfies all conditions for a Group except associativity
But what is the identity?
Jul
28
comment Subgroups of a cyclic group and their order.
@user1729: Suppose that $H$ is a subgroup of order $d$, where $d \mid n$. Now $H = \langle a^k \rangle$ for some $k \mid n$. Then $a^k$ has order $n/k$, but on the other hand $a^k$ must also have order $d$. Hence $k = n/d$.
Jul
26
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Jul
25
awarded  Good Answer
Jul
25
awarded  Mortarboard
Jul
25
awarded  Nice Answer
Jul
25
answered Prove that $\sqrt 2 + \sqrt 3$ is irrational
Jul
23
answered Is $G$ a semidirect product of $Z(G)$ and $\operatorname{Inn}(G)$?
Jul
19
answered Normal subgroup of direct product of two groups
Jul
16
comment Is a finite group determined by the family of all its 2-generated subgroups?
For the $Sub_1$ case, consider two distinct $p$-groups of same order, of exponent $p$ (with $p$ odd). For example, let $G_1$ be the elementary abelian group of order $p^3$ and $G_2$ the Heisenberg group of order $p^3$. So if $Sub_1(G_1)$ and $Sub_1(G_2)$ are isomorphic, it does not necessarily follow that $G_1$ and $G_2$ are isomorphic.
Jul
11
answered If $G$ is a finite group of order $n$, why is it isomorphic to its centralizer in $S_n$?
Jul
4
comment Probability that two randomly chosen permutations will generate $S_n$.
For references, see A001691 and A071605 from OEIS. To me it seems that Q2 is hard, at least if you want exact values. I think you can figure out bounds from the many papers that study the probability of generating $S_n$ with a pair of random permutations. For a recent result in this direction and some history about the problem, the paper "A. Maroti and C. M. Tamburini, Bounds for the probability of generating the symmetric and alternating groups, Arch. Math. (Basel), 96 (2011), 115-121" looked interesting.
Jun
18
comment Are Clifford groups very *non-commutative*?
Do you mean $ab = gba$ so $g = [a,b]$?
Jun
15
accepted If $(|G|, |H|) > 1$, does it follow that $\operatorname{Aut}(G \times H) \neq \operatorname{Aut}(G) \times \operatorname{Aut}(H)$?
Jun
15
comment Intersection of the $p$-sylow and $q$-sylow subgroups of group $G$
Because $P, P_1, P_2, \ldots, P_n$ are distinct, you have $P \cap P_i = 1$. Note that $(A \times B) \cap (C \times D) = A \cap C \times B \cap D$ and similarly for cartesian products with more than two factors.