All Questions
7 questions
1
vote
1
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314
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Small order examples of non-nilpotent finite groups in which every minimal normal subgroup intersects the center nontrivially
I know that $p$-groups have the property, and all nilpotent groups also do. I wanted to have some non-nilpotent examples. Then I checked quasisimple groups, and found every proper normal subgroups of ...
user517681
0
votes
1
answer
78
views
Let $G = \{A \in G L_n(\mathbb{R}) : \mathrm{det} A = 2^k$ for some $k \in Z \}$. Show that $G$ is a normal subgroup of $H = GL_n(\mathbb{R})$.
Let $G = \{A \in G L_n(\mathbb{R}) : \mathrm{det} A = 2^k$ for some $k \in Z \}$. Show that $G$ is a normal subgroup of $H = GL_n(\mathbb{R})$.
I want to show that for $g \in G$; $gHg^{-1} = H$
But ...
1
vote
1
answer
84
views
How to prove that the following subgroup is normal?
Let $M(n,\mathbb{R}$) denote the set of all $n \times n$ matrices with real entries (identified with $\mathbb{R}^{n^2}$ and endowed with its usual topology) and let $GL(n, \mathbb{R})$ denote the ...
- 4,592
7
votes
3
answers
4k
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Commutator Group of $\operatorname{GL}_2(\mathbb{R})$ is $\operatorname{SL}_2(\mathbb{R})$
Let $\operatorname{GL}_2(\mathbb{R})$ be the general linear group of $2\times2$ matrices and $\operatorname{SL}_2(\mathbb{R})$ the special linear group of $2 \times 2$ matrices. Show that the ...
- 6,773
1
vote
1
answer
869
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$SO(3)$ and normal subgroups
As the title says i have to find wheter the subgroup $H$ of $SO(3)$, defined by:
$\omega\in H, \omega=
\begin{pmatrix}
1 & 0 \\
0 & \lambda \in SO(2)
\end{pmatrix}
\space (1)$
The ...
2
votes
1
answer
190
views
Does $GL_2(\Bbb Z_3)$ have a normal subgroup of order 8, or 16?
So I know any group of order $48$ has a normal subgroup of order $8$ or a normal subgroup of order $16$ (or possibly both).
I am trying to expand upon this topic, figuring which of these conditions ...
- 49
3
votes
1
answer
2k
views
Subgroups of special linear group SL$(n, \mathbb{Z})$
Are there characterizations of subgroups of a special linear group SL$(n, \mathbb{Z})$?
Since SL$(n, \mathbb{Z})$ has infinite order, it would be enough if I know how to generate subgroups of SL$(n, \...
- 53