# Questions tagged [group-theory]

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 is the study of groups.

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### Help with disjoint cycle decomposition and finding image/pre-image of a S8 group

i have this exercise and i'm confused about the part where i find the image and pre-image. I solved it for 5 in alfpha, found the image and preimage (preimage is number that 5 results from and image ...
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### Finding a Group isomorphism G

According to wiki, Isomorphism: Given two groups $(G, ∗)$ and $(H, {\displaystyle \odot })$, a group isomorphism from $(G, ∗)$ to $(H, {\displaystyle \odot })$ is a bijective group homomorphism ...
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### Cycle type of a permutation in $S_n$ and its relation to partition of $n$ and its Young diagram

I know that it's possible to assign to each permutation its cycle type. I found two definitions of the cycle type and its relation to a partition of $n$: First definition Given $\sigma \in S_n$ ...
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### Beautiful Frieze Groups [closed]

Maybe this is more of an art question than a math question but here it goes: What are the most beautiful themed (all having similar style) examples of all the 7 Frieze groups that you've seen? Here'...
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### Unique cyclic subgroup in dihedral group is characteristic [duplicate]

I have seen a proof of "the cyclic subgroup H of dihedral group $D_{n}$ where n>=3 is characteristic". In that proof a result was used. " H is the unique cyclic subgroup generated by an element of ...
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### Reduced multiplicative residue modulo p [duplicate]

I would like if someone could provide or show me where I can find the proof that $\mathbb{Z} _n^*$ is cyclic when n is prime. In particular I'm after a simple proof that involves fermats little ...
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### Coordinate Changes in Group Presentations [on hold]

Let $G$ be a finitely presented group with presentation $\langle x_0, \dots, x_n \; | \; R(x_0) \rangle$. Let $f$ be an automorphism of $G$ such that $f(x_0), x_1, \dots x_n$ generates $G$. Is it true ...
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### Dimension of pro-$p$ group

Problem. Let $G$ be a pro-$p$ group of finite rank. Prove that $$\mathrm{dim}(G) = \lim_{k \to \infty}\frac{\log_{p}|G:G^{p^k}|}{k}.$$ By definition $$\mathrm{dim}(G) = \mathrm{d}(H)$$ where $H$ is ...
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### Why does cocycle condition imply a group?

In group cohomology, the 2-cocycle condition emerges from associativity (see e.g. here). From the answer to a previous question we see (at least for normalized cocycles) that the cocycle condition ...
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### Find group of automorphisms

In group $GL(2,R)$ I have subgroup H, generated by 2 elements: $a=\begin{pmatrix} 1 & 2 \\ 0 & 1\\ \end{pmatrix}$ , $b=\begin{pmatrix} 1 & 3 \\ 0 & 1\\ \end{pmatrix}$ I ...
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### Proving $G\simeq G^{op}$ as categories

Exercise 1.2.23 https://arxiv.org/pdf/1612.09375.pdf Let $G$ be a group, regarded as a one-object category all of whose maps are isomorphisms. Then its opposite $G^{op}$ is also a one-object ...
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### When the sum of two permutations is a permutation? [closed]

I have to work with encrypted data, a problem arises as follows: Assume that we have $\sigma_1, \sigma_2 \in S_X$, where we consider the set $X$ as the abelian group $Z_n$. Each element of $X$ is an ...
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### Subgroup of order 2 of a group of order 56 [duplicate]

I was given the following question: Does every group of order 56 contain a subgroup of order 2. I know that the Sylow theorems guarantee the existence of an order 8 subgroup. Is there a general ...
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### Suppose $AN_1=AN_2=G$, $A\cap N_1=A\cap N_2=1$ and $N_1,N_2$ are normal in $G$, do we have $N_1\cong N_2$?

Let $G$ be a finite group. Suppose $AN_1=AN_2=G$, $A$ is a subgroup of $G$, $A\cap N_1=A\cap N_2=1$ and $N_1,N_2$ are normal in $G$, do we have $N_1\cong N_2$? I think this is not true,but I failed ...
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### A condition for normalized 2-cocycles from the existence of the inverse element in a group extension

It's a well-known fact that if $A$ an Abelian group and $G$ is a group, then all group extensions of $G$ by $A$ is isomorphic with the group ($A\times G,\,\bullet)$, where the group operation $\bullet$...
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### A functor $\mathcal G\to \mathbf{Set}$ is the same as a left $G$-set

I'm trying to understand the first part of Example 1.2.8 from here: https://arxiv.org/pdf/1612.09375.pdf Let $Ob(\mathcal G)=\{\star\}$. A functor $F:\mathcal G\to \mathbf{Set}$ consists of: An ...
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### Uniform powerful pro-$p$ group isomorphic to $Z_{p}^{d}$

Problem. Let $G$ be a uniform pro-$p$ group and suppose that $G$ has an abelian open normal subgroup. Show that $G/Z(G)$ is finite, deduce that in fact $G \simeq \mathbb{Z}_{p}^{d}$ for some $d$. (...
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### Is $\mathbb{Z}_{4} \oplus \mathbb{Z}_{6} \cong \mathbb{Z}_{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}$?

Is $\mathbb{Z}_{4} \oplus \mathbb{Z}_{6} \cong \mathbb{Z}_{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}$? In my opinion, this statement is correct because the maximal order of element in each ...
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### Is there a group $G$ such that $\mathrm{Aut}(G)\simeq (\mathbb{Q},+)$? [duplicate]

My guess is that no such $G$ exists. All I know is that if $\mathrm{Aut}(G)$ is abelian, then $G$ is two-step nilpotent. Since $\mathbb{Q}$ is locally cyclic (i.e. every f.g. subgroup is cyclic), can ...
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### Free Groups and Actions on Trees

Here is the proof of a theorem I am working through in Geometric Group Theory by Clara Loh: The first paragraph shows that if $F$ is free, then it admits a free action on a (non-empty) tree. This ...
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### Finding basis for a representation of $D_8$.

Let $G=D_8=\langle a,b\mid a^4=b^2=1,b^{-1}ab=a^{-1}\rangle$. The character table of $D_8$ is known and is Let $U:=\bigg\{\sum\limits_{1\leq i<j\leq 4} a_{ij}x_ix_j\mid a_{ij}\in\mathbb{C}\bigg\}$ ...
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### Finite Sylow subgroups of periodic groups

Let $G$ be a (possibly infinite) periodic group, and suppose that $G$ admits a maximal finite $p$-subgroup $P$. By this I mean that we do not assume that $P$ is not strictly contained in any other $p$-...
### What does 'every monomorphism in $R$-$\mathsf{Mod}$ is a kernel' mean?
In Algebra: Chapter $0$ by Aluffi, at the end of page 161, he writes: Since every submodule $N$ is then the kernel of the canonical projection $M \to M/N$, our recurring slogan becomes, in the ...
Let $\psi:{H}\mapsto{H}'$ be a surjective homomorphism. Can there exist a proper subgroup ${M}$ of ${H}$ such that $\psi({M})={H}'$? Can there exist more than one such subgroup? Should they be ...