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PhD Thesis. - University of Birmingham

A 3-local Characterization **of** theThompson Sporadic Simple GroupbyRachel Ann Abbott FowlerA thesis submitted toThe **University** **of** **Birmingham**for the degree **of**Doctor **of** PhilosophySchool **of** MathematicsThe **University** **of** **Birmingham**February 2007

- Page 2 and 3: AbstractIn this thesis we character
- Page 4 and 5: ContentsIntroduction 10.1 Overview
- Page 6 and 7: IntroductionThe Classification of F
- Page 8 and 9: (iii) G β = N G (Z β ) ∼ 3 1+2+
- Page 10 and 11: Show:For γ ∈ Θ β define P γ =
- Page 12 and 13: (i) H/F = K, where K ∼ = 3 3 : Sy
- Page 14: We can consider F as a GF(2)K-modul
- Page 17 and 18: Proof. See [13, Theorem 2.2.3].□L
- Page 19 and 20: The following lemma shows that the
- Page 21 and 22: C Q (r). Next we show that φ is su
- Page 23 and 24: Then (1.1) is said to be a series o
- Page 25 and 26: Lemma 1.2.3 Suppose that Q is an ex
- Page 27 and 28: Proof. If C P (x) = 1, then C P (x)
- Page 29 and 30: Lemma 1.3.2 Let W be a GF(2)〈x〉
- Page 31 and 32: ⎧⎛⎞⎫⎪⎨⎜where δ = −
- Page 33 and 34: Definition 1.4.3 Let A = A(A 1 , A
- Page 35 and 36: Parts (iii) and (iv) of Lemma 1.5.3
- Page 37 and 38: that Syl 3 (L αβ ) ⊆ Syl 3 (L
- Page 39 and 40: G α•.• G βL α•G αβ•
- Page 41 and 42: that G γ /〈X Gγ 〉 is not a 3
- Page 43 and 44: Chapter 2A Recognition Result for G
- Page 45 and 46: and therefore |Z(S)| ≥ 3 3 , whic
- Page 47 and 48: where a, b, c are elements of the f
- Page 49 and 50: (ii) G ∼ = M c L;(iii) G ∼ = Co
- Page 51 and 52: Chapter 3Some Strong Closure Result
- Page 53 and 54:
Type 3 These contains two 1-dimensi

- Page 55 and 56:
subgroup of L of index 2. So V ⊳

- Page 57 and 58:
Lemma 3.2.2 Suppose that X ∼ = Al

- Page 59 and 60:
We conclude this section with some

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Now suppose that F is cyclic. Hence

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(ii) Let a = (123), b = (123)(456)(

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(a) Alt(8);(b) Sym(7);(c) (3 × Alt

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The Sylow 3-subgroups have order 3

- Page 69 and 70:
have that F R/R is a 2-subgroup of

- Page 71 and 72:
Let R ≤ T ∈ Syl 2 (N G ((R ∩

- Page 73 and 74:
y Goldschmidt’s Theorem, see Theo

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(vi) |Q α ∩ Q β | = 3 4 .Proof.

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(iii) If z ∈ S αβ has order 3,

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G α•.• G βL α•G αβ•

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g ∈ L α \N Lα (S αβ ) and def

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elementary abelian. HenceΦ(W β )

- Page 85 and 86:
Chapter 5Proof of Theorem AIn this

- Page 87 and 88:
Remark 5.1.2 We note that given any

- Page 89 and 90:
Γ(γ).Let ψ be the action of G γ

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Definition 5.1.9 Let γ ∈ Θ β a

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the above and hence (iii) follows.

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By Lemma 5.1.11 (ii), P δ /W δ∼

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centralizes a 1-space. We see that

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Proof. Suppose that W β W β+2 /

- Page 101 and 102:
Proof. We have that G is a 3-genera

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Chapter 6Proof of Theorem BThis cha

- Page 105 and 106:
5.1.8, N γ∼ = Q8 and C Mγ (t γ

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Proof. By Lemma 6.1.5, C G (Y ) ∩

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the subgroups of CgS αβ(Y )/Z β

- Page 111 and 112:
Lemma 6.2.4 NeG (J)/C eG (J) ∼ =

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(ii) Y has six conjugates in ˜G β

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Let R ∈ I ∗eG (J, 3′ ). So, J

- Page 117 and 118:
So, a given conjugate of Y is conta

- Page 119 and 120:
Now suppose that C F (A 2 ) = C F (

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C Z/Z(E) (B) = {0} and so C E (B) =

- Page 123 and 124:
Proof. We have that C G (R) = CeG (

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We show that we only need to consid

- Page 127 and 128:
First suppose case 1 occurs. If x

- Page 129 and 130:
Concluding RemarksWe recall that th

- Page 131 and 132:
Bibliography[1] M. Aschbacher. GF(2

- Page 133:
[25] I.A. Korchagina, C.W. Parker,