Nicky Hekster
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 Aug 27 comment Quotient Group question ambiguity about group order First of all, the binary operation of the group is inherited by the quotient group, but then applied to the "multiplication" of cosets. Your second confusion: you should make a difference between a multiplicative and additive notation of the binary group operation. In multiplicative format $g^k$ Is similar to the additive $g + \dots + g$ ($k$ times). Aug 27 comment Quotient Group question ambiguity about group order You are mixing up the concepts of the cardinality (~order) of the coset itself with the order of the coset as a group element in the quotient group. Aug 25 comment Intersection of subgroup of finite index with infinite subgroup is infinite $f_1=f_2h$ implies $f_2^{-1}f_1=h \in F \cap H$. Hence$f_1(F \cap H)=f_2(F \cap H)$. Aug 25 answered Intersection of subgroup of finite index with infinite subgroup is infinite Aug 25 revised Determine number of elements of order 12 of a group added 290 characters in body Aug 24 comment Determine number of elements of order 12 of a group You should now be able to find out that yourself. One of the coordinates should have order $3$, and another order $4$. So figure out how to construct those elements. Aug 24 answered Determine number of elements of order 12 of a group Aug 24 answered p-element centralizing a Sylow p-subgroup Aug 23 comment Standard notation for indices in group theory? The books of the group theory authorities (B. Huppert, I.M. Isaacs, D.J.S. Robinson, J. Rose) all use the last one. The straight lines reflect the cardinality of a set, in this case the number of cosets. Similarly, $|G|$ is used for the order (cardinality) of the group $G$. I myself sometimes use another one: index$[G:H]$ too. I certainly would not use the first one with the round brackets. But that is maybe the French school against the Anglo-Saxon conventions. Aug 22 comment assume $M/N$ be a chief factor of $G$. Why $M/N$ has prime order or order $4$? Why do you keep asking and changing the same questions? Aug 21 answered Let $H/N$ is nilpotent quotient subgroup. Then is $N$ characteristic in $H$? If no, then what condition need? Aug 21 revised Problems with P. Hall theorem proof (The problem involves the use of Frattini's argument) added 8 characters in body Aug 21 answered Problems with P. Hall theorem proof (The problem involves the use of Frattini's argument) Aug 21 comment Find the class equation for the following groups Can you show how you got these answers, which are by the way correct? Aug 20 answered If a group $G$ is not simple does it follow that it is isomorphic to the direct product of two nontrivial groups? Aug 20 comment A group of order 2p (p prime) and other conditions - prove abelian. @Ilya.K. You should know that $G/Z(G)$ is cyclic implies $G$ is abelian. Maybe you have not seen this "theorem". Aug 16 answered Prove $[N(H):H]\equiv [G:H](\mod p)$ Aug 16 comment Given 3 spheres, find the equation of the plane that touches each of the spheres on the same side..? Excellent hint! Aug 15 comment Group $G$ acting on $\Omega$ such that each $\alpha \in \Omega$ has unique $p$-element fixing $\alpha$. Excellent Derek, +1 from me! Aug 14 answered Prove that Euler phi function is multiplicative by a given theorem