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Math Help - Various Group Theory questions

  1. #1
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    Various Group Theory questions

    Hey guys I'm struggling a bit here

    1. Let G be a finite group.

    a) Suppose that H is a subgroup of G. What is meant by |G : H|, the index of H in G.

    b) Suppose that x ∈ G. Describe the subset of G known as the conjugacy class of x.

    c) Suppose that x ∈ G. Describe the subgroup CG(x) known as the centralizer of x.

    d) State the formula which relates the size of the conjugacy class of x ∈ G to its centralizer CG(x) .

    e) Show that if the finite group G has exactly two conjugacy classes then it must be cyclic order 2.



    2. Let V,W be vector spaces over the field F and let σ: V → W be a linear transformation.

    a) Define the rank and nulity of σ

    b) Prove that σ is injective if and only if nulity of σ = 0

    c) Prove that if dimW < dimV then σ cannot be injective


    Any help would be greatly appreciated! Thanks
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  2. #2
    Senior Member abhishekkgp's Avatar
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    Re: Various Group Theory questions

    Quote Originally Posted by rgjf1307 View Post
    Hey guys I'm struggling a bit here

    1. Let G be a finite group.

    a) Suppose that H is a subgroup of G. What is meant by |G : H|, the index of H in G.

    b) Suppose that x ∈ G. Describe the subset of G known as the conjugacy class of x.

    c) Suppose that x ∈ G. Describe the subgroup CG(x) known as the centralizer of x.

    d) State the formula which relates the size of the conjugacy class of x ∈ G to its centralizer CG(x) .

    e) Show that if the finite group G has exactly two conjugacy classes then it must be cyclic order 2.



    2. Let V,W be vector spaces over the field F and let σ: V → W be a linear transformation.

    a) Define the rank and nulity of σ

    b) Prove that σ is injective if and only if nulity of σ = 0

    c) Prove that if dimW < dimV then σ cannot be injective


    Any help would be greatly appreciated! Thanks
    I can help you with the group theory questions. All those questions are asking just the definition of various concepts(except for the last one). you can get that straight from any text book. Still i will define them here:
    a) define the set K=\{gH|g \in G \}. Then |G:H|=|K|.

    b) define M= \{gxg^{-1}|g \in G \}. M is the conjugacy class of x.

    c) C_G(x)=\{g \in G|gxg^{-1}=x \}.

    d) |G:C_G(x)|= the size of conjugacy class of x in G.

    e)use the CLASS EQUATION.
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  3. #3
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    Re: Various Group Theory questions

    Thanks a lot Abhishek, that's some really useful information! Do you have any idea about the vector space problem? I have a definition of the rank and nulity but can't for the life of me see how that can prove the mapping is injective...
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  4. #4
    Senior Member abhishekkgp's Avatar
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    Re: Various Group Theory questions

    Quote Originally Posted by rgjf1307 View Post
    Thanks a lot Abhishek, that's some really useful information! Do you have any idea about the vector space problem? I have a definition of the rank and nulity but can't for the life of me see how that can prove the mapping is injective...
    i have not read linear algebra so i can't help you on that. wish i could.
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  5. #5
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    Re: Various Group Theory questions

    "injective" means that if σ(a)= σ(b) then a= b. nullity of a linear transformation is the dimension of its null space, the subspace of vectors, v, such that σ(v)= 0. Saying the nullity of σ is 0 means that σ(v)= 0 only for v= 0.

    You are asked to "Prove that σ is injective if and only if nulity of σ = 0".
    "if and only if" means you need to different proofs.

    1) Suppose that σ is injective and prove its nullity is 0. Indirect proof: assume that the nullity of σ is not 0. That means that the null space contains non-zero vectors: there exist some vector a such that σ(a)= 0. But for any linear transformation, σ, it is true that σ(0)= 0.

    2) Suppose that nullity of σ= 0 and prove σ is injective. Indirect proof: assume that σ is not injective. That means that there exist vectors, a and b, such that a\ne b but σ(a)= σ(b). What can you say about σ(a- b)?
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