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kakarukeys
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Can isomorphism of matrix groups
[tex]\phi: G_1 \rightarrow G_2[/tex]
always be expressed by
[tex]\phi(M) = S M S^{-1}[/tex]?
[tex]\phi: G_1 \rightarrow G_2[/tex]
always be expressed by
[tex]\phi(M) = S M S^{-1}[/tex]?
cathalcummins said:Right, this thread is the closest to the topic I could find.
My question is more convention choice than anything else.
The finite Cyclic [tex]C_3[/tex] group is defined by:
[tex]{e,c,b(=c^2)}[/tex]
where [tex]e[/tex] is the identity, [tex]c[/tex] is rotation through [tex]\frac{2\pi}{3}[/tex] etc. I'm keeping formalities to a minimum here.
Clearly, we are rotating a triangle with directed sides in one plane through three angles, yeah?
Now, my question has to do with representations of this group.
Let me begin with how I am learning about groups.
The definition I'm working from (and taking the example from) is in line with H F Jones' "Groups, Representations and Physics Second Ed".
A representation of a Group [tex]G[/tex] is the pair [tex]{\iota, V}[/tex] where [tex]V[/tex] is the vector space which is also a group and
[tex]\iota : G \mapsto V[/tex]
is a homomorphism. With the usual condition that [tex]\iota[/tex] preserves the group structure.
So, some books understandably skip the generality of [tex]V[/tex] and claim that [tex]V=M_{n \times n}[/tex] the set of invertible nxn matrices.
And to represent [tex]C_3[/tex] we will use the finite subspace
[tex]R(\theta)= \begin{bmatrix}\cos \theta & -\sin \theta & {0} \\ \sin \theta & \cos \theta & {0} \\ {0} & {0} & {1}\end{bmatrix} [/tex]
where [tex]\theta=0, \frac{2 \pi}{3},\frac{4\pi}{3}[/tex]
A representation of [tex]C_3[/tex] is the pair [tex]{\iota, M}[/tex] where [tex]M[/tex] is the vector space with three elements
[tex]M=R(0),R(\frac{2 \pi}{3}),R(\frac{4\pi}{3})[/tex]
[tex]\iota : G \mapsto M[/tex]
Now, finally, I can ask my question.
According to the book [tex]R(\frac{2 \pi}{3})[/tex] would be the "representation" of [tex]c[/tex] in [tex]C_3[/tex]. I find this kind of confusing.
So what would ([tex]\textbf{x'},\textbf{x}[/tex] are just Cartesian column vectors)
[tex]\textbf{x'}=R(\theta)\textbf{x}[/tex]
be?
Can someone shed some light on how to view this definition as intuitive. I understand that a representation shouldn't demand a co-od system. I suppose, to me, it just seems like we're identifying an 'operator' ([tex]R(\theta)\in M_{n \times n}[/tex]) with a 'state' ([tex]e,c,b \in C_3[/tex]).
cathalcummins said:May I redefine in more familiar language(to my course).
A representation of C_3 is the pair {i,R^2}, where
i : C_3 -> R^2
is the function taking the following form:
i(g) = R(2pi/3)
where, g \in C_3 and R is the 2X2 rotation matrix.
Isomorphism in matrix groups refers to a one-to-one mapping between two groups of matrices that preserves the structure and operations of the groups. This means that the elements of the two groups are matched in a way that preserves the multiplication and inverse operations between them.
To prove that two matrix groups are isomorphic, you must show that there exists a bijective mapping between the elements of the two groups, and that this mapping preserves the group structure and operations. This can be shown through various methods such as finding a matrix representation of the groups and comparing their properties, or showing that the groups have the same order and structure.
No, not all matrix groups are isomorphic. Isomorphism depends on the structure and operations of the groups, and not all groups have the same structure or operations. Some groups may have different orders, while others may have different properties that prevent them from being isomorphic.
Isomorphism in matrix groups allows us to study and compare different groups of matrices by finding similarities in their structures and operations. It also helps us understand the properties and behaviors of one group by studying the properties of another isomorphic group.
Yes, isomorphism is a concept that can be applied to various mathematical structures, not just matrix groups. It can be used to compare and analyze groups, rings, fields, and other algebraic structures by establishing a one-to-one mapping that preserves their operations and properties.