# Questions tagged [derived-subgroup]

Also called the commutator subgroup of a group is the subgroup generated by all commutator elements of that group. Should be used with the (group-theory) tag.

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### Commutator Subgroup is a Subgroup of "$G^2$"

Show that the commutator subgroup $G'$ of a group $G$ is a subgroup of $G^2$ defined by $G^2=\{x^2:x \in G\}$. I've tried writing the general element of $G'$; $hgh^{-1}g^{-1}$ as the square of some ...
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### $R$ points of derived subgroup of algebraic group

I'm reading through Milne's book on algebraic groups, and in corollary 6.19 he writes: Assume that $G$ is affine or smooth, then (c) for all $k-$algebras $R$, $(\mathcal{D}G)(R)$ consists of the ...
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### The only proper normal subgroups of a nonabelian quasisimple group are subgroups of its centre.

This is part of a bunch of exercises set by my academic supervisor. As such, I'm not sure whether all the hypotheses are needed for the conclusion. Please note that hints are preferred. The Question: ...
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### Prove/disprove: Let $G$ s.t $|G| = p^3 \implies |G'| \leq p$. [duplicate]

Let $G$ s.t $|G| = p^3$ for some prime $p$. Prove or disprove: $|G'| \leq p$. I couldn't think of any counter-examples so I started proving this and I'm stuck unfortunately and was hoping to seek ...
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### Solvable Groups and Derived Subgroups. Why does $D(G_i) \subset G_{i + 1} \implies D_i(G) \subset G_i$.

I'm working through the proof of the following theorem: $G$ is a solvable group $\Leftrightarrow$ There exists $n$ such that $D_n(G) = {1}$, where $D_i(G)$ is the $i$th derived subgroup of $G$. Proof: ...
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### Request vetting of understanding of problem on commutator subgroups.

Let there be dihedral group $D_n$ and take two elements with first ($a$) being a generator rotation and other ($b$) any reflection. Which group is $[D_n,D_n]$? Till now not covered quotient groups, ...
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### $\{ a \mid a\not \in [G,G],\ a \in Z(G) \} \cup \{e\}$ is a group or not

Let $G$ be a group. I want to check whether the set $$\{ a \mid a\not \in [G,G],\ a \in Z(G) \} \cup \{e\}$$ is a subgroup or not, where $[G,G]$ and $Z(G)$ are the commutator subgroup and center of ...
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### Quick question about metabelian group

The following classic exercise question: show that the group $G$ is metabelian iff $G'' = e$ ($G''$ denotes $(G')'$, the commutator subgroup of the commutator subgroup of $G$). I know that the ...
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### A problem in group theory and representation theory.

Suppose that $H = \{1, h, h^2, \ldots, h^{n-1}\}$ is a normal subgroup of a finite non-abelian group $G$ having order $n$. It is known that $H$ is cyclic with generator $h$. Let $c_G$ be the number ...
1 vote
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### Show that the twisted commutator, $T_g = \{x \in G \mid \exists k\in\Bbb Z, xgx^{-1} = g^k\}\le G$ for some $g \in G$ of finite order

I'm studying for qualifying exams, and this problem has me stumped. It should be straightforward (the definitions are fundamental), but I can't figure out the trick. Let $G$ be any group and let $g$ ...
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### Is there an easy way to find the derived group of $S_5$?

In order to find the derived group of $S_5$ I've tried using Lagrange’s Theorem to find the order of the possible subgroups but $O(S_5)=2^3\cdot 3 \cdot 5$ so there are too many possible subgroups to ...
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### How to find the derived subgroup of $Q_8$

Given the group $Q_8=\{ \pm1, \pm i, \pm j, \pm k \}$ is there an easy way to find its derived subgroup? I've tried by hand and this seems such a big task to do manually. After this, by thinking a ...
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Given a group $(G, \cdot)$ is there a way to find its derived subgroup other than calculating it by hand element by element? For instance, if a group is abelian you know that its derived subgroup is $\... 1 vote 0 answers 39 views ### Given$H \lhd G$and$G/H$is abelian then the derived group$G'$or$[G,G]$satisfies$G'\subseteq H$Given$H \lhd G$and$G/H$is abelian then the derived group$G'$or$[G,G]$satisfies$G'\subseteq H$I've approached this problem the following way: As$H$is abelian then given$a,b \in G$,$$\... 3 votes 0 answers 48 views ### Intersection of the derived subgroup with a cyclic group Let$G$be a group and$G'$be the derived subgroup. Let$C=\langle c \rangle $be a cyclic subgroup of$G$. Is it true that the intersection$H=G'\cap C$is trivial? I was thinking that if$g\in H$, ... 4 votes 1 answer 465 views ### Let$G$be a group,$H\unlhd G$, prove that the commutator subgroup$H'$of$H$is a normal subgroup in$G$. Let$G$be a group,$H$, a normal subgroup of$G$, prove that the commutator subgroup$H'$of$H$is a normal subgroup in$G$. My ideas: I want to prove it like this:$gH'g^{-1} = H'\forall g \in ... 1 vote
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### If every finitely generated subgroup of a linear group G is solvable, then G is solvable

I've been working on this question for a long time now and I still have no idea how to approach it. I tried induction on the dimension of the matrices in G but can't complete the induction step. Any ...
1 vote
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### The relationship between the commutator subgroup and the center of a group

I just started reading about the upper and lower central series of a group. I'm wondering if there is a general relationship between the commutator subgroup $G'$ and the center $Z(G)$ of a group $G$. ...
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### If $G$ nilpotent and $G/G'$ is cyclic then $G$ is cyclic? [duplicate]

If $G$ nilpotent and $G/G'$ is cyclic then $G$ is cyclic? It is very easy to see that this is true when $G$ is finite: If $G$ is finite nilpotent then all maximal subgroups are normal, so $G/N$ has to ...
1 vote
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### why "this means that $G'' \subset Z(G')$"?

I am trying to understand the proof of the following question: Show that if $G'/G^{''}$ and $G^{''}/G^{'''}$ are both cyclic then $G^{''} = G^{'''}.$[you may assume $G^{'''} = 1.$ Then $G/G^{''}$ acts ...
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### Why $G'' \triangleleft G$?

Here is the question I want to solve: Show that if $G'/G''$ and $G''/G'''$ are both cyclic then $G'' = G'''$. [you may assume $G''' = 1$. Then $G/G''$ acts by conjugation on the cyclic group $G''$. ...
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### Information about $P'$ in $P \rtimes Q$

Let $p$ and $q$ be distinct primes. Suppose we have a group $G = P \rtimes Q,$ where $P$ is a non-abelian $p$-group and $Q$ is an abelian $q$-group such that it is a subgroup of ${\rm Aut}(P).$ Then ...
1 vote
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### Non-existence of perfect groups with order $180$

A group $G$ is perfect when $G=G'$. $\textbf{Question.}$ Prove that there are no perfect groups of order $180$. $\textbf{My attempt.}$ I assumed there is such a $G$, so it is not solvable. I was ...
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### Let $H\leq G$ s.t. whenever 2 elements of $G$ are conjugate then the conjugating element can be chosen from within $H$. Prove that $G'\subseteq H$.

Question: Let $H\leq G$ such that whenever two elements of $G$ are conjugate, then the conjugating element can be chosen from within $H$. Prove that $G'\subseteq H$. Solution: So, if $g_1$ and $g_2$ ...
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1 vote
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### The commutator subgroup of a nonabelian simple group $G$ is $G$ itself [duplicate]

I'm studying elementary level of algebra and I'm trying to prove that the commutator subgroup of a nonabelian simple group is the original group itself. It is trivially false if the group is abelian, ...
1 vote
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### Proves about $L_n$ (lower central series of a group $G$)

I'm trying to solve this problem from my abstract algebra course: The lower central series of a group $G$ is defined by means of $L_0=G$, $L_{n+1}=[G,Ln]=\langle[g,h]:g\in G,h\in L_n\rangle$, for any ...
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### Let $G$ be a group. Prove that $G'$ is central iff ${\rm Inn}(G)$ is abelian.

This is Exercise 4.11 of Roman's "Fundamentals of Group Theory: An Advanced Approach." According to this search and Approach0, it is new to MSE. The Details: On page 33 of Roman's book, ...
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### Prove that $D'_n=\langle x^2\rangle$

I want to check if my solution to one problem from my group theory course is valid. The problem is: Given $D_n=\{x^iy^j:0\leq i<n,0\leq j<2\}$, prove that $D'_n=\langle x^2\rangle$. My attempt ...
1 vote
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### If $G$ is group with $|G|=p^3$, $p$ prime, then $G'=Z(G)$

I'm trying to solve this problem from my abstract algebra text book: Being $G$ a non-abelian group of order $p^3$, with $p$ prime. Prove that $G'=Z(G)$ In my notation $G'$ is the derived subgroup of ...
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### Assume $G$ is a finite group such that every maximal subgroup of $G$ is normal in $G$ and for $H \leq G$ we have that $HG’=G$,then show that $H=G$.

Assume $G$ is a finite group such that every maximal subgroup of $G$ is normal in $G$ and for $H \leq G$ we have that $HG’=G$,then show that $H=G$. Where $G^\prime$ is the commutator subgroup (AKA ... ### Condition that shows when there are elements in $[G,G]$ which are not commutator.(Looking for a proof)
It's known that for a given group $G$ and the derived subset $[G,G]$ it's not true that every element of $[G,G]$ is a commutator. I've seen a condition that shows when there are elements in $[G,G]$ ... 