# Questions tagged [group-theory]

For questions about the study of algebraic structures consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility.

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### Transfer and fusion in a centralizer

Suppose $G$ is a finite group of order divisible by $8$, with an element $\tau$ of order 2 whose centralizer $C_G(\tau)$ is elementary abelian of order 4. I suspect $G/[G,G]$ must have even order, but ...
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### What is the amalgamated free product $\mathbb{Z} \ast_\mathbb{Z} \mathbb{Z}$? [duplicate]

This is a small exercise to help with my understanding of amalgamated free product of groups. I think I got it partially, but not sure if I can/should further simplify it. The theoretical construction ...
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### cyclic group under multiplication

𝑁=𝑁(𝑍𝑛) for set of all nilpotent elements in Zn and 1+𝑁={ 1 + z | z ∈ N} I am to prove that 1+𝑁(Z(27)) is a cyclic group under multiplication. I am aware that a group is cyclic if there exits ...
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### Is $H$ is a subgroup of $D_4?$ Yes/No [duplicate]

Is $H=\{ x\in D_{4} \mid x^2=1\}$ is a subgroup of $D_4?$ My attempt : I think not Take the elements $s$ and $rs$ of $D_4$ Here $s^2=1$ and $( rs)^2 =rsr^{-1}s=s^2=1$ But $s(rs)=rs^2=r \neq 1$ ...
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### A problem about the orders of a finite group [duplicate]

Can I prove that a finite group with more than one element has element with prime order, like this, for some element "$a$" let the order of it be $n$ ($a$ is not the identity) let $n=pk$, ...
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### Co-cyclic group and abelian monolithic group

A $\textbf{cocyclic group}$ is a group $G$ with $\displaystyle\bigcap_{\{1\}\neq H\le G}H \neq\{1\}$. By this referement: https://encyclopediaofmath.org/wiki/Cocyclic_group result equivalent this ...
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### Exercise 1.1.7 of Dummit and foote . Is $G$ a group?

This question is from "Abstract Algebra" by Dummit and Foote , third edition , page 21 , exercise 1.1.7 . Let $G = \{ x \in \mathbb R | 0 \leq x < 1 \}$ and for $x,y \in G$ let $x*y$ be ...
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### Find all $7$-Sylow subgroups of $S_{14}$

Recently, while studying for my group theory class, I tried to solve the following problem Find all $7$-Sylow subgroups of $S_{14}$. Since the order of $S_{14}$ is $14!$ I know that $P$, a $7$-Sylow ...
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### Prove the subset of a linear code consisting of codewords with even weight is a subgroup.

"Let $C$ be a linear code over $Z_2$. Let $C^+$ be the subset of $C$ consisting of those elements of $C$ with even weight. Show that $C^+$ is an additive subgroup of $C$." It's an exercise ...
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### Artin's theorem 2.33: subgroups of $\mathbb{Z}$

Theorem 2.33 in Artin states: Let $S$ be a subgroup of the additive group $\mathbb{Z}^+$. Either $S$ is the trivial subgroup $\{0\}$, or else it has the form $\mathbb{Z}a$, where $a$ is the smallest ...
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### No nonzero integers > 1 or < -1 have a multiplicative inverse. How does one prove this formally?

I understand intuitively why this is true: no integers > 1 or < -1 have a multiplicative inverse. However, I'm not sure how to prove this formally. Without relying on intuition, how do you KNOW ...
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### Are there any interesting theorems of first-order group theory

I've heard that theoremhood in first-order group theory is uncomputable, so there should be some theorems that are difficult to prove. However the few theorems of group theory I know are about ...
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### Can elements of automorphism groups be expressed as some form of matrix?

I was reading on automorphism groups and Galois theory and this idea came to mind: Since by definition, automorphisms are isomorphisms, we have $\phi(a*b)=\phi(a)\cdot\phi(b)$ where $\phi$ is an ...
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### Prove or disprove $\langle a,b,c\rangle$ is free

Suppose that $\langle a,b\rangle$, $\langle b,c\rangle$, and $\langle a,c\rangle$ are free, is it true then that $\langle a,b,c\rangle$ is a free group of rank 3? We assume that $a,b,c$ are distinct ...
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### Let $G$ be a group and $H\unlhd G$ such that $[G:H]=20$ and $|H|=7$. Suppose $x\in G$ and $x^7=e$. Show that $x\in H$.

I'm currently on an exercise problem from Dan Saracino Abstract Algebra, Exercise 11.20. It says the following: Let $G$ be a group and $H$ be a normal subgroup of $G$ such that $[G:H]=20$ and $|H|=7$. ...
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### Let $G$ be a finite group and $Z(G):=\{x\in G : gx=xg \ \ \forall g\in G\}.$ Prove that $[G:Z(G)]$ is not a prime number. [duplicate]

Let $G$ be a finite group and $Z(G):=\{x\in G : gx=xg \ \ \forall g\in G\}.$ Prove that $[G:Z(G)]$ is not a prime number. I think that I should use the Lagrange Theorem, i.e., If $Z(G)$ is a subgroup ...
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### On the proportion of 2-groups.

Suppose $T(n)$ denotes the number of non-isomorphic 2-groups (groups of order a power of 2) of order at most $n$, and $G(n)$ the number of non-isomorphic finite groups of order at most $n$. It is well ...
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### $(G/N)/\ker\pi \cong G/H\to\ker\pi＝H/N$
Let $G$ be a group and $N$ and $H$ are normal subgroup of $G$ and $N$ is normal subgroup of $H$. $\pi:G/N\to G/H$ be natural projection, that is, $x\pmod{N}\to x\pmod{H}$. Then, I would like to ...