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Mathematics
Linear and Abstract Algebra
Uniqueness of Group Presentations
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[QUOTE="fresh_42, post: 6219848, member: 572553"] A group representation of a group ##G## is a group homomorphism ##\varphi\, : \,G \longrightarrow \operatorname{Aut}(H)## into an automorphism group of some set ##H##, which can be another group, or a vector space, in which we call the representation linear and the automorphism group ##\operatorname{GL}(V)##. This a group representation is the triple ##(G,\varphi,H)##. It makes no sense to speak of uniqueness, since ##\varphi## is what makes it unique. Different homomorphisms mean different operations mean different representations. The same goes for minimality. We can always define ##g.h :=h## i.e. map every element on the identity automorphism. This is automatically minimal in which sense you ever measure size. Since ##\varphi## is required to be a group homomorphism, this is equivalent to the invariance of relations. One has to prove that ##\varphi## is a group homomorphism, so depending on how a representation is defined, the properties have to be proven, i.e. calculated. [/QUOTE]
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Mathematics
Linear and Abstract Algebra
Uniqueness of Group Presentations
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