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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define the set of all affine real-valued functions

Define the set of all affine real-valued functions G := { $f$_a,b : a,b ∈ $R$ , a≠0}, where $f$_a,b : $R$ → $R$, x → ax + b. This is a group under composition ○. a) N := ...
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2answers
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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 ...
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1answer
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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 ...
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1answer
40 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 ...
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1answer
37 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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0answers
28 views

isomorphism and subgroups

If we have two finite groups like automorphism groups of vector spaces, in order to check whether two finite groups are isomorph or not, how can we use subgroups of them?
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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 ...
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1answer
63 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, ...
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175 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: $\mathbb{R}/\mathbb{Z} \cong \mathcal{C}$, ...
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1answer
140 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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2answers
241 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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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), ...
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1answer
79 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 ...
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1answer
83 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 ...
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3answers
170 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?
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1answer
163 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 ...
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4answers
438 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 ...
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2answers
108 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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391 views

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

In the tradition of this question, why is $\operatorname{PGL}(2,4)\cong A_5$?
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2answers
844 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!
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7answers
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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, ...