Isomorphisms between members of infinite families of mathematical objects (finite simple groups, Lie groups etc), that are not examples of a pattern of such isomorphisms.

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2
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1answer
44 views

Isomorphism between $E$ and $Lin(E)$ : infinite dimensional case.

Does $E$ and $Lin(E)$ where $Lin(E)=\{A : E \rightarrow E ∣ A \quad\text{is linear}\}$ are isomorph if $E$ infinite dimensional case ? I know that if $E$ is finite dimension the result is true ...
0
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1answer
38 views

Discrete math Group - Isomorphism and Automorphism

Let G be a Cyclic group Prove or disprove: A.let $ a,b \in G \quad $ so the function $ f:G \to G,f(a^k) = b^k$ is Automorphism of G(which means G is Isomorphism to herself) B.let a,b generators ...
-1
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2answers
49 views

isomorphic quotient rings?

I have trouble in determining, whether two rings are isomorphic: Let's have $R = GF(3)$ and rings $R[x]/(x^2+x+2)$ and $R[x]/(x^2+2x+2)$. How can one determine whether these two rings are ...
1
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1answer
25 views

Are there infinite positive $n \in \Bbb N$ such that ($D^+_n$, |) is isomorphic to ($D^+_{4100}$, |)?

Are there infinite positive $n \in \Bbb N$ such that ($D^+_n$, |) is isomorphic to ($D^+_{4100}$, |)? I'm guessing no because I can't relate every element of ($D^+_{4100}$, |) to ($D^+_n$, |) because ...
0
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1answer
45 views

How can I prove that $(ℤ, ≺)$ is not isomorphic to $(ℕ, ≤)$

We define the relation $≺$ between pairs of integers like this: $n≺m$ is true if and only if one of the following conditions holds: a) $0≤n≤m$ b) $0≤n$ and $m<0$ c) $n<0 , m<0$ and ...
0
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1answer
53 views

Abstract Algebra: Find a number of non-isomorphic subgroup of $S_3$

I'm having difficulty in understanding the method to find the solution for this question. I repeat Question: Find the number of non-isomorphic subgroup of $S_3$ So is this the way to find the ...
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2answers
116 views

Prove that $PSL(2,9)\cong A_6.$

I have tried to solve the Exercise 8.12, page 227, in Joseph J. Rotman, An Introduction to the Theory of Groups, Fourth Edition, Graduate texts in Mathematics, Springer-Verlag, New York (1995). The ...
0
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1answer
68 views

Question regarding isomorphisms in low rank Lie algebras

I am reading Brian Hall's book 'Lie Groups, Lie Algebras, & Representations' and on p.271 I find that in low rank Lie algberas there are some isomorphisms. For example, ...
4
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2answers
231 views

Examples of non-obvious isomorphisms following from the first isomorphism theorem

I am learning the first isomorphism theorem, and I am working with some isomorphisms to practice for my upcoming test. I know some of the basic ones like: $\mathbb{R}/\mathbb{Z} \cong \mathcal{C}$, ...
6
votes
1answer
153 views

Why is $PGL_2(5)\cong S_5$?

Why is $PGL_2(5)\cong S_5$? And is there a set of 5 elements on which $PGL_2(5)$ acts?
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votes
2answers
293 views

Binary tetrahedral group and $\rm{SL}_2(\mathbb F_3)$

The binary tetrahedral group $\mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup $$ \mathbb T = \left\{ \pm 1, \pm i, \pm j, \pm k, \dfrac{\pm 1 \pm i \pm ...
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votes
2answers
204 views

How to count the conjugates of an exotic $S_5$?

It can be read off the The Elliott configuration - a $5$-coloring of $K6$ - that $S_6$ has an exotic $S_5$ subgroup (it's not a point stabilizer) which I will call $X_5 = \langle (1\;3\;6\;4\;5), ...
2
votes
1answer
81 views

Give an isomorphism between the linear spaces $U\times V$ and $L(U,V)$

I was hoping someone could help me with the above question (give an isomorphism between the linear spaces $U\times V$ and $L(U,V)$. I have a hunch that I should work with the bases of U and V but the ...
0
votes
1answer
85 views

In $D_4$, find $H_1$ isomorphic to $\mathbb{Z}_4$ and $H_2$ isomorphic to the Klein Four Group with $D_4/H_1$ isomorphic to $D_4/H_2$.

a) In $D_4$, find $H_1$ isomorphic to $\mathbb{Z}_4$ and $H_2$ isomorphic to $V$, the Klein Four group, with $D_4/H_1$ isomorphic to $D_4/H_2$. b) In $D_4$, find subgroups $H$ and $K$ with $H$ normal ...
0
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3answers
175 views

Why are these elements generators of cyclic groups?

My example says: AutC$_4$, $C_4 = \{1, a, a^2, a^3\}$ And then it points to $a$ and $a^3$ and says these are generators and these are the two elements of order 4. Why is this?
2
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1answer
169 views

Why is $\operatorname{Spin}(3) \cong \operatorname{SU}(2)$?

Why is $\operatorname{Spin}(3) \cong \operatorname{SU}(2)$? I can't seem to realize $\operatorname{Spin}(3)$ explicitly as a matrix group and so I'm having issues constructing an isomorphism, of ...
10
votes
4answers
507 views

Elementary isomorphism between $\operatorname{PSL}(2,5)$ and $A_5$

At this Wikipedia page it is claimed that to construct an isomorphism between $\operatorname{PSL}(2,5)$ and $A_5$, "one needs to consider" $\operatorname{PSL}(2,5)$ as a Galois group of a Galois cover ...
9
votes
2answers
112 views

Isomorphism between $\mathfrak o(4,\mathbb R)$ and $\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $

I've been trying to find a Lie algebra isomorphism $$\mathfrak o(4,\mathbb R)\cong\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $$ but haven't managed so far. I have written down the ...
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2answers
419 views

Why is $PGL(2,4)$ isomorphic to $A_5$

In the tradition of this question, why is $\operatorname{PGL}(2,4)\cong A_5$?
10
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2answers
940 views

Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?

Could you please explain me the reason why they are isomorphic? Thanks, bye!
15
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7answers
1k views

Why $PSL_3(\mathbb F_2)\cong PSL_2(\mathbb F_7)$?

Why are groups $PSL_3(\mathbb{F}_2)$ and $PSL_2(\mathbb{F}_7)$ isomorphic? Update. There is a group-theoretic proof (see answer). But is there any geometric proof? Or some proof using octonions, ...