# 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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### Derived series of a square-free order group stabilizes

I have been studying from these notes on groups, rings and fields by Lenstra and I find myself struggling with problem 1.20 which states the following Let $G$ be a finite group of squarefree order. ...
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### Does there exist an element of order $4$ in $GL_2(\mathbb{Z})/GL_2(\mathbb{Z})'$?

For a group $G$, let $G'$ denote the commutator of the group $G$, and if $H \leq G$ the left cosets will be denoted as $gH$. Now, I understand the fact that $[SL_2(\mathbb{Z}):GL_2(\mathbb{Z})'] = 2.$ ...
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### Proving the isomorphism type of the commutator subgroup of the dihedral group $D_n$

For any $n$, to what group is the commutator subgroup of the dihedral group $D_n$ isomorphic to? My solution is below. I request verification, feedback, and improvements. In particular, can you help ...
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### When is the special linear group $SL_n(\Bbb F_q)$ over a finite field $\Bbb F_q$ solvable?

Recall that the special linear group $$SL_n(\Bbb F_q)=\{ A\in GL_n(\Bbb F_q)\mid \det(A)=1\},$$ where $\Bbb F_q$ is the field of $q$ elements for finite $q$. A group $G$ is solvable if the derived ...
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### If $G^{(i)}$ denotes the ith derived subgroup of $G$ then $G^{(i+1)}$ is a proper subgroup of $G^{(i)}$

Is this statement true? If $G^{(i)}$ denotes the ith derived subgroup of $G$ then $G^{(i+1)}$ is a proper subgroup of $G^{(i)}$ for all $i$. I think this statement is false by the symmetric group. ...
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### If $G/G’$ is cyclic, prove by induction that $G/Z(G)$ is cyclic if $G$ is $p$ group. [duplicate]

Let $G$ be a $p$ group. If $G/G’$ is cyclic, prove $G/Z(G)$ is cyclic by induction. I tried playing with the base cases : if $|G|=p$ or $p^2$, $G$ is abelian, so $G/Z(G)=G/G =\{G\}$ which is ...
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### Testing if an element is in the derived subgroup

Let $G$ be a finite group. Let $G':=[G,G]$ be the derived subgroup. How do you test if an element $g \in G$ is in $G'$? I'm specifically interested in how to do this in GAP. In other words I ...
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### What is known about this generalisation of the derived group?

Recently I've been revising a course on Representation Theory. As part of that course, we ended up proving that the intersection of the kernels of all 1D representations of a group is its derived ...
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### The commutator of Holomorph of generalized quaternion is abelian?

Let $Q_{2^{n}} = \langle x, y | x^{2^{n-1}}=y^4 = 1, x^{2^{n-2}}=y^2, y^{-1}xy = x^{-1} \rangle$ - generalized quaternion group of order $2^{n}$. $\operatorname{Hol}(Q_{2^{n}})$ - Holomorph of this ...
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### If $F$ is free and $R$ is normal in $F$, then $F/R'$ is torsion-free, where $R'=[R,R]$

If $F$ is free and $R$ is normal in $F$, then $F/R'$ is torsion-free, where $R'=[R,R]$. This is Exercise 11.50 in Rotman's An Introduction to the Theory of Groups with the following hint attributed ...
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### In $G/K(G)$ for commutator group $K(G)$, it is said that $[a][b]=[a][b][b^{-1}a^{-1}ba]=[a][b][b^{-1}][a][b][a]=[b][a].$ Why the first equality?

Let $K(G)$ be the commutator subgroup. It is said that in the quotient space $G/K(G)$ \begin{align} [a][b]&=[a][b][b^{-1}a^{-1}ba]\\ &=[a][b][b^{-1}][a][b][a]\\ &=[b][a]. \end{align} ...
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### Let $H = \langle X \rangle$ then $H' = \langle [x_1, x_2]^h \mid x_1, x_2 \in X, h \in H \rangle$ [duplicate]

Let $H = \langle X \rangle$, then $$H' = \langle [x_1, x_2]^h \mid x_1, x_2 \in X, h \in H \rangle.$$ Obviously $$\langle [x_1, x_2]^h \mid x_1, x_2 \in X, h \in H \rangle \leq H'.$$ Any tips on how ...
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### 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 $a,b \in G$, \...
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### 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 $g\in H$, ...
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### The commutator of the product of 2 groups

Let $G=H\times A$ be the internal product of $A$ and $H$, I proved that if "$A$ is abelian then $G′=H′$ " But if we were not given that $A$ is abelian is $G′=H′\times A′$ ?
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### Show ${\rm SL}(n,q)$ is perfect for odd prime power $q>3$.

This is Exercise 3.2.10 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to this search and Approach0, it is new to MSE. The Details: On page 8, ibid., Let $R$ ...
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### Is it "obvious" that nested commutators generate the derived series?

The derived series of a group is constructed iteratively, taking repeated commutator subgroups. A commutator subgroup is famously not only the set of commutators but the group they generate. This ...
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### Number of conjugacy classes of $G/[G,G]$

Let $G$ be a finite group. We can define a relation $R$ on $G$ by setting $aRb \iff b = g^{-1} a g$ for some $g \in G$. The conjugacy class of an element $a \in G$ is defined as its equivalence class: ...
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### What is the number of degree $1$ representations of a finite group?

Let $G$ be a finite group and $G'$ be the derived subgroup of $G.$ Then I know that every degree $1$ representation of $G$ factors through $G/G'.$ How does it imply that the number of degree $1$ ...
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I am trying to understand an argument in a group theory, which is not my strong suit (I mostly work with semigroups and rarely delve into the actual structure of groups). Let $\boldsymbol{\mathcal{V}}$...