# 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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### Conjugate diagonal matrices

Let $M$ be the set of matrices that have precisely one entry in each row/column that is nonzero and $D$ be the set of invertible diagonal matrices ( so all entries down the diagonal are nonzero). ...
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### Hom$_{Set}(G \times H,R) \cong$Hom$_{Set}(G,R)\otimes$ Hom$_{Set}(H,R)$?

If $G$ and $H$ are finite groups and $R$ is a ring does Hom$_{Set}(G \times H,R) \cong$Hom$_{Set}(G,R)\otimes$ Hom$_{Set}(H,R)$ if Hom sets have the usual R-algebra structure?
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### $G$ is direct product of $H_1,\ldots,H_k$

Let $G$ be a direct product of the groups $H_1,\ldots,H_k$, let $K\vartriangleleft G$. Prove that if $Z(G)$ doesn't contain $K$, then there is $i$ such thet $K\cap H_i \neq\{ e \}$ ($e$ is the ...
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### confusion about cosets and quotient space

The set of cosets $V/U = \{ v+U: u \in V \}$ with operations $$(v+U) + (w+U) = v+w+U$$ $$a(v+U) = av+U$$ (which is well defined) is a vector space called Quotient Space. I am having a difficult ...
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### How can I understand $\Bbb Z\times \Bbb Z/2\Bbb Z$

This may be stupid request, but I would like to have a intuition for the group $\Bbb Z\times \Bbb Z/2\Bbb Z$ in terms of 'real' objects. 'Real' could mean geometric but not necessarily. I perhaps what ...
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### Show the map from the product of disjoint subgroups of G to G is an isomorphism [duplicate]

Question: Show that if $G$ is a group with two normal subgroups $H$ and $K$ such that $G=HK$ and $H\cap K=\{e\}$, then the map $(h,k)\mapsto hk$ is an isomorphism of groups from $H\times K$ to $G$ ...
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### Find a homomorphism of general linear group whose kernel consists of matrices with positive determinant

Let $G=GL_n(\mathbb{R})$, the group of invertible $n\times n$ real matrices, Let $N$ be the subset of $G$ given by $\{A\in G:\det(A)>0\}$ Let $H=\{1,-1\}$ considered as a group under ...
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### Find all the $p$-Sylow subgroups of $D_6$.

$|D_6|=12=2^23.$ I started with $3$. I know that the number of $3$-Sylow subgroups, denoted $n_3$, is: $1,4,7...$ and I also know that $n_3|2^2$. e.g, $n_3=1, 4$. How can I show that it can't be $4$? ...
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### let $G$ to be group such that $O(G)=p^2$ where $p$ is prime,prove that $G$ is cyclic or $G$ is Direct product of two cyclic subgrops of order $n$. [duplicate]

the only hint that i got is Sylow's first theorem, which implies that if $p^n$ is any prime power dividing $O(G)$, then $G$ has a subgroup of order $p^n$. in our case $p$ devides $p^2$, then we can ...
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### How to implement a group action in Sage ( for educational purposes )?

Let S= {"A","B","C","D"} and S4= SymmetricGroup(4). I want to create a table of the action S4 x S -> S which standardly permutes the letters in the set. The table should look like: ...
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### Subgroups of Semidirect Product of the elementary abelian group of order 8 by $S_3$

What are the subgroups of the semidirect product of the elementary abelian group of order 8 by $S_3$? This is the group $(\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2)\rtimes S_3$ of order 48; ...
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### If a group has one p-sylow subgroups, then this subgroup must be normal.

I've learned that this is true. Why, basically? I'd appreciate you help.
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### Canonical form by J. A Green

Let $f(x)=x^d-a_{d-1}x^{d-1}\cdots-a_0$ be a irreducible polynomial over finite field $\mathbb{F}_q$, $C(f)$ be its company matrix. Let $A$ be a matrix with minimum polynomial $m(x)$. For the sake of ...
This post is a sequel of: Which finite groups have faithful complex irreducible representations? A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation. ...