Tagged Questions

46 views

Is S a group under matrix addition

Another matrix question! Let $$S=\{A \in M_2(\mathbb{R}):f(A)=0\}\text{ and }f\left(\begin{bmatrix}a&b\\c&d \end{bmatrix}\right)=b$$ Is S a group under matrix addition. Either prove that ...
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Please check my proof on: $\sim$ is an equivalence relation $\Leftrightarrow S<G$

Problem: Let $\emptyset\ne S\subset G$, where $G$ is a group, and define a relation on $G$ by $a\sim b\Leftrightarrow ab^{-1}\in S$. Show that $\sim$ is an equivalence relation if and only if $S$ is a ...
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Free action by cyclic group.

Let $G$ be a group acting on a set $X$. If $g\in G$ has no fixed points, prove or disprove the cyclic group $\left \langle g \right \rangle$ acts freely on $X$. edit: Can also assume $g$ has finite ...
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What is M-bar in factor groups? [duplicate]

If G is a group and N $\triangleleft$ G, show that if $\bar{M}$ is a subgroup of G/N and M = {a $\in$ G | Na $\in$ $\bar{M}$}, then M is a subgroup of G, and M $\supset$ N. If $\bar{M}$ is normal in ...
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For a discrete abelian group $G$, is the Gelfand Representation of $\ell^1(G)$ injective?

Given a discrete group $G$, we can consider the Banach $*$-algebra $\ell^1(G)$, with convolution product $(\xi*\eta)(g)=\sum_{h\in G}\xi(h)\eta(h^{-1}g)$ and involution ...
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Existence of an element in a group of certain order if an element of other order exists

Show that if a group $G$ of order $1089=3^2\cdot 11^2$ contains an element of order $9$ then it also contains an element of order $33$. I tried to see what would Sylow theorems tell for this problem ...
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Total number of non isomorphic groups of order 122.

Let $G$ be group of order $122 = 61 \cdot 2 = p \cdot q$ , where $p < q$ are primes. I know that there exists a unique non abelian group of order $pq$ and one abelian non isomorphic group of order ...
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Show that: $\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$

Show that: $$\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$$ where $D_n$ is dihedral group and $a$ is generator of order $n$.
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isomorphism of algebric torus

I'm trying to prove the following: Let $D_n=(\mathbb{C}^{\times})^n$ (an algebric torus of rank $n$). Assuming $D_k$ is isomorphic to $D_n$ as an algebric group. Prove that $k=n$. So far, I managed ...
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Automorphism problem

For arbitrary group $(G,\cdot)$ let $Aut(G) =$ {$f: G \to G | f$ is a isomorphism} be set of all automorphisms of group $G$. We assume that $(Aut(G),\circ)$ where $\circ$ is addition of mappings is ...
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Subgroups of $(\mathbb Z_n,+)$

The problem is to define all subgroups of $(\mathbb Z_n,+), n \in \mathbb N$. My guess is if n is prime number, then there is only trivial subgroups. If n is not prime, then I can factorize it, and ...
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$eH \in G/H$ is the only element with finite order

$\newcommand{\ord}{\text{ord}}$This is a question in my book: $G$ is an infinite Abelian group and $H=\{ a \in G \mid \ord(a) < \infty \}$ is a normal subgroup of $G$, show that $eH$ is the ...
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Minimal non solvable group is simple

I suppose to prove in my homework that Every minimal non solvable group is simple. I can't find the way. Thank you.
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On modular group and quadratic forms

Let $\Gamma$ be the modular group, is the group of linear transformations of the upper half of the complex plane. Let $\mathbb Q_{N^2{d_K}}/\Gamma$ (the group of positive definite primitive quadratic ...
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How many normal subgroups does a non-abelian group $G$ of order $21$

How many normal subgroups does a non-abelian group $G$ of order $21$ have other than the identity subgroup $\{e\}$ and $G$? 1) 0 2) 1 3) 3 4) 7 I think option 1 is incorrect because every group ...
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Particular nontrivial group has a nonidentity automorphism [duplicate]

If $G$ is a nontrivial group that is not cyclic of order 2, then $G$ has a nonidentity automorphism. This is the exercise of hungerford algebra in the chapter $IV$ MODULES. Can you help me please?
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Calculate all the local automorphisms

The Kohn - Nirenberg domain $\Omega_{KN}$ defined by $$\Omega_{KN}=\left\{(z,w)\in \Bbb C^2:\text{Re}\ w+|zw|^2+|z|^8+\dfrac{15}{7}|z|^2\text{Re}\ z^6<0\right\}$$ How to compute all the ...
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How can there exist an isomorphism between this group and the cyclic group $(\mathbb Z,+)$?

I have a group over $\mathbb Z$, defined by the binary operation $*$, such that $a*b:=a+b+2$. From the previous exercise, I have deduced that the identity-element is $-2$ and that it is an abelian ...
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Chain condition (ACC and DCC ) for a group

If $G$ is a group that satisfies both chain conditions (ACC & DCC ) and there exists a group $H$ with $G×G≅H×H$, we can say $G≅H$? If G satisfies both chain conditions and K and H groups satisfies ...
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Subgroups of $F^*$ are cyclic

Q: If $F$ a field then every finite subgroup of $F^*$ is cyclic. Solution: Suppose $d\mid |G|$ for $G$ subgroup of $F^*$ and $G$ not cyclic. Suppose $A,B$ subgroups of $G$ of order $d$. Then ...
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What is the orbit of a set?

I have the following question: Let $G = S_4$. What is $\textrm{Orb}(H)$ (under $G$) when $H =V_4$, $H = \textrm{Sym} \{1,2,3\}, H = <(1234)>$ ($G$ is acting by conjugation)? I don't quite ...
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Algebra I: Cyclic Generators

The direct product $\mathbb{Z}_{45} \times \mathbb{Z}_{98}$ is cyclic and isomorphic to $\mathbb{Z}_{4410}$ because $gcd(45,98)=1$; furthermore the element $n=([1]_{45},[1]_{98})$ is a cyclic ...
All permutations from $S_6$ and $S_7$ by which $(1,2)(3,4,5)$ is conjugate to itself
The task is to find all permutation $\tau$ from $S_6$ and $S_7$ such that: $$\tau^{-1}(12)(345)\tau=(12)(345)$$ I think the answer is: $\{id \in S_6 , id \in S_7 , (67) \in S_7\}$ I would just like ...