Tagged Questions

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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If $C_G(x) \leq H$ for every $p$-element $x \in H$, then $p$ cannot divide both $|H|$ and $|G:H|$

This is problem 1.D.2 in Isaacs, Finite Group Theory. I am self-studying, so would appreciate a proof verification. Note: in this book, all groups are assumed finite unless otherwise stated. Fix ...
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Without using the Jordan-Hölder theorem, show the following.

I am not sure how to prove the following. I am practicing for an exam in a few months time, and found this from a past paper in my university which unfortunately doesn't come with solutions. $G$ ...
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Application of Burnside's theorem? Composition factors of $|G|=p^2q$ are…

Continuing for my study on practicing group theory... I am now stuck on this problem about composition factors, $G$ is a group such that $|G|=p^2q$ where $p \neq q$ and $p,q$ are prime. Prove that ...
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$\sigma \big( (a,1),(1,b) \big) = \big( (a^3,1),(a^2,b) \big)$ is an automorphism [on hold]

How to show that the map $\sigma \big( (a,1),(1,b) \big) = \big( (a^3,1),(a^2,b) \big)$ is an automorphism? Thank You.
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Difficulty understanding subgroups for certain simple groups

Firstly, I am studying at the high-school level so please excuse my lack of understanding of these concepts. Consider the group $\Bbb C_{3v}$ of symmetries of an equilateral triangular lamina. It ...
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Prove that conjugacy classes of Symmetric and Alternating Groups are equal, when…

I've been working with this problem for my own exercise, as I haven't touched on Group theory for quite a while. But I am too incompetent to give it a proof. I've searched for answers but ...
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Showing an isomorphism exists between two groups

For $a, b ∈ \Bbb Z$, let $B(a, b) ∈ M(2, \Bbb Z)$ be defined by $$B(a, b) = \begin{bmatrix} a & 3b \\ b & a \\ \end{bmatrix}$$ Let $S = \{B(a, b) | a, b ∈ \Bbb Z\} ⊆ M(2,\Bbb Z)$. Show that ...
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Can I place the rubiks3cube pieces in the distorted position I intend to get?

Right now I am trying to get the distorted position like this: in each face only one diagonal is solved and no similar colour is on a face other than the diagonal pieces mentioned previously. For ...
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Stabilizer Conjugation

This may be a straightforward question, but if I have a group $G$ acting on a set $A$, and two elements $a,b\in A$ belong to the same orbit, how do I show that their stabilizers are conjugate. So ...
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Exercise from Artin's Algebra. Describe the centralizer $Z (\sigma)$ of the permutation $\sigma = (153)(246)$ in the symmetric group $S_{7}$, and compute the orders of $Z (\sigma)$ and of $C ... 0answers 16 views Unit group over a real closed field Let$F$be a real closed field. Let${\mathbb{S}^1}(F)$denote the abelian group$(S^1(F),*)$where$S^1(F) = \{(a,b) \in F^2 \ | \ a^2 + b^2 = 1\}$with the law$(a,b)*(c,d) = (ac - bd,ad + bc)$. ... 1answer 32 views Is the group of automorphisms of an abelian group itself abelian ?? [on hold] I think the statement is true . But I have no idea how to prove it right. 0answers 17 views If$P \in \mbox{Syl}_p(G)$is abelian, then$G$has normal$p$-complement if$P \le H$and$H$has one and controls$p$-transfer Let$G$be a finite group, and denote by$A^p(G)$the smallest normal subgroup such that$G / A^p(G)$is an abelian$p$-group. Suppose$P \le H \le G$for a fixed Sylow$p$-subgroup$P$of$G$. We say ... 0answers 13 views Extending a homomorphism from a subgroup to whole group where the target is not a divisible group I was reading this post of stack exchange. So in the question if the circle group is replaced by$\mu_{p-1}$which is the group of$(p-1)^{th}$root of unity and if the group$G/H$is assumed to a ... 1answer 15 views Some properties of the largest abelian$p$-factor group and its kernel$A^p(G)$. Let$G$be a finite group and let$A^p(G)$be the unique smallest normal subgroup of$G$for which the corresponding factor group is an abelian$p$-group (that$A^p(G)$is well-defined is an immediate ... 1answer 34 views Number of group homomorphism from$Z_8$⊕$Z_2$to$Z_4$⊕$Z_4$I know that there does not exist an isomorphism from$Z_8$⊕$Z_2$to$Z_4$⊕$Z_4$as there exist an element of order 8 in$Z_8$⊕$Z_2$and no element of order 8 in$Z_4$⊕$Z_4$. But what about ... 1answer 56 views Reference Request for Topics in Group Theory I'm taking an honors-level algebra class and we're getting into group actions, Sylow subgroups and semidirect products. These topics are fairly intimidating, but the lengthier discussions in the book ... 0answers 57 views The order of$ab$when$a,b$commute Let$a,b$be two group elements of finite order that commute. I thought that$|ab| = \text{lcm}(|a|,|b|)$. My proof was that$(ab)^n = a^n b^n =e$if and only if$a^n = b^n = e$if and only if ... 2answers 19 views If$\phi$is a homomorphism between cyclic groups$G$and$F$, why must$|\phi(G)|$divide$|G|$? I am reading a solution to another problem that counts the homomorphisms. I don't get why$|G|/|\phi(G)|$, I get that$|F|/|\phi(G)|$by Lagrange's theorem. I guess it has to do with if$|G| = m$, ... 1answer 26 views Confused about this exercise question: if a set with a certain binary operation is a group I tried to answer the following exercise: Let$S$be a nonempty set with an associative operation that is left and right cancellative. Assume that for every$a$in$S$the set$\{a^n \mid n=1,2,3, ...
As we all know that in group $S_n$ every pair of distinct disjoint cycles commute .my doubt is is it reverse all true,mean if a pair of distinct cycles commute ,then they have to be disjoint??.i ...