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1 Groups
Theorem 1.1 (Lagrange). Let G be a finite group and let H be a subgroup of G. Then |H| divides |G|.
Theorem 1.2. If H is a subgroup of G then for all a, b ∈ G |aH| = |bH| and either aH = bH or aH ∩ bH = ∅ (i.e. the cosets partition the group)
Theorem 1.3 (Cayley’s Theorem). Every finite group G is isomorphic to a subgroup of a permutation group. If G has order n then it is isomorphic to a subgroup of the symmetric group Sn.
Theorem 1.4. Let (S, ◦) be a G-Set (i.e. ◦ is a G-action on S). If s inS and Hs is the stabilizer of s while Os ⊆ S is the orbit of S then there is a bijection ϕ : G/H → Os
defined by aH → as. Further ϕ(gC) = g ◦ ϕ(C) for every coset C and g ∈ G.
Corollary 1.5 (Counting Theorem). Let (S, ◦) be a G-Set. If s ∈ S then we have (order G) = (order of stabilizer of s)(order of orbit of s)
|G| = |Hs||Os|
or equivalently |Os| = [G : Hs]
Corollary 1.6. |S| = σi|Oi| where each orbit Oi occurs exactly once.
Theorem 1.7. Conjugation is a group action
Lemma 1.8. Let |G| = pe. Then the center of G has order > 1
1.1 Sylow’s Theorems
Theorem 1.9 (1st Sylow Theorem). Let G be a finite group of order n (|G| = n) and let p be a prime such that
- n = pe^ · m
- p does not divide m
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then there is an element of G of order pe.
Corollary 1.10. IF p divides |G| then G contains an element of order p.
Theorem 1.11 (2nd Sylow Theorem). Let K be a subgroup of G with order divisible by p. Let H be a p-Sylow subgroup of G. Then there is a conjugacy subgroup of H′^ = gHg−^1 such that K ∩ H′^ is a Sylow p-subgroup of K
Lemma 1.12. Let (S, ◦) be a G-Set and let s ∈ S. Let s′^ be in the ordit of s, say s′^ = a ◦ s. Then Gs′^ = aGsa−^1
Where Gs, Gs′ are the stabilizers of s, s′^ respectively.
Corollary 1.13. The Sylow p-subgroups of a group are all conjugate and every conjugate of a Sylow p-subgroup is a Sylow p-subgroup.
Theorem 1.14 (3rd Sylow Theorem). Let |G| = n = pem where p does not divide m. Let s be the number of p-Sylow subgroups of G. Then s divides m and s = ap + 1 for some integer a.
1.2 Abelian Groups
Theorem 1.15. Let A be an abelian group. Then {a ∈ A : (∃n ∈ Z)na = 0} is a subgroup of A, called the Torsion Subgroup of A
Recall that we say F = 〈v 1 , · · · , vn〉 if every element of F can be expressed as a linear combination (over Z) of v 1 ,... , vn in a unique way. We can think of {v 1 , · · · , vn} as a “basis” for F over Z.
Theorem 1.16. Let F be a free abelian group of rank r and let G be a subgroup of F of rank s ≤ r (but s > 0 ). Then G is a free abelian group of rank s ≤ r. Further F has a set of generators 〈u 1 , · · · ur〉 such that G is generated by
v 1 = a 11 u 1 + a 12 u 2 + · · · + a 1 rur v 2 = a 22 u 2 + · · · + a 2 rur .. .
vs = assus + · · · + asrur
for some aij such that all aii are positive
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Theorem 2.2. Let V be a finite dimensional vector space over a field F and let B is a basis for V. Then the map 〈v, w〉 = vtB AwB is a bilinear form for every matirx A (where vB , wB are as in the previous theorem)
Lets now deal with Real Vector Spaces
Theorem 2.3. Let A be the matrix associated to a bilinear form 〈, 〉 on V relative to a basis B. Then those matrixes which represent the same form relative to different basis are those matrixes of the form
QAQt
for some Q ∈ GLn(F ).
Lemma 2.4. If you change basis by an orthogonal change of base (i.e. the change of base matrix is orthogonal) then the dot product is preserved.
Lemma 2.5. The matrixes which represent the dot product under some basis are those of the form P P t^ for P ∈ GLn(R).
Lemma 2.6. A bilinear form is symmetric if and only if the matrix associ- ated to it is symmetric.
Lemma 2.7. Let B be an orthonormal basis for V relative to 〈, 〉. Then the matrix associated to 〈, 〉 under B is the identity matrix.
Theorem 2.8 (Gram-Schmidt). Let 〈, 〉 be a positive definite symmetric bi- linear form on a finite dimensional vector space V. Then there is an or- thonormal basis for V
Theorem 2.9. The following are equivalent
- A represents the dot product
- There is a P ∈ GLn(R) such that A = P P t
- A is symmetric and positive definite.
Theorem 2.10. Let V, 〈, 〉 be a Euclidean space (where |v| =
〈v, v〉). Then we have the Schwartz inequality|〈v, w〉| ≤ |v| · |w| |v + w| ≤ |v| + |w|
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Lemma 2.11. Let V be a vector space and let 〈, 〉 be a bilinear form on V. If W ⊆ V is a subspace then 〈, 〉 restricts to a bilinear form 〈, 〉|W on W. Further if 〈, 〉 is positive definite or symmetric so is 〈, 〉|W
Lemma 2.12. Let 〈, 〉 be a non-identically zero symmetric bilinear form on V. Then there is a v ∈ V such that 〈v, v〉 6 = 0.
Theorem 2.13. Let 〈, 〉 be a symmetric bilinear form on V (a finite dimen- sional vector space). If W ⊆ V is a subspace such that 〈, 〉|W , the bilinear form restricted to W , is non-degenerate, then
V = W ⊕ W ⊥
Theorem 2.14. Let 〈, 〉 be a symmetric bilinear form on V (a finite dimen- sional vector space). Then there is an orthogonal basis for V.
Theorem 2.15 (Sylvester’s Law). The signature of a symmetric bilinear form on a finite dimensional vector space is independent of the basis.
Theorem 2.16. Let 〈w 1 ,... , wr〉 be an orthonormal basis for W ⊆ V (rela- tive to a bilinear form 〈, 〉). Then the orthogonal projection πW (v) of v onto W is the vector
πW (v) = 〈v, w 1 〉w 1 + · · · + 〈v, wn〉wn
Corollary 2.17. Let 〈v 1 ,... , vn〉 be an orthonormal basis for V (relative to a bilinear form 〈, 〉). Then
(∀v ∈ V )v = 〈v, v 1 〉v 1 + · · · 〈v, vn〉, vn
2.1 Hermitian Forms
Theorem 2.18. Let V be a finite dimensional complex vector space and let 〈, 〉 be a hermitian form on V. If B is a basis for V then there is a hermitian matrix A such that 〈v, w〉 = v∗ B AwB where vB , wB are the matirx representations of v, w with respect to the basis B
Theorem 2.19. Let V be a finite dimensional complex vector space and let B is a basis for V. Then the map 〈v, w〉 = vtB AwB is a hermitian form for every hermitian matirx A (where vB , wB are as in the previous theorem)
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3 Rings
Theorem 3.1 (Substitution Principle). Let ϕ : R → R′^ be a ring homomor- phism
(a) Given an element α ∈ R′^ there is a unique homomorphism Φ : R[x] → R′^ which agrees with the map ϕ on constant polynomials and sends x → α.
(b) Given elements α 1 , · · · , αn ∈ R′^ there is a unique ring homomorphism Φ : R[x 1 ,... , xn] such that Φ|R = φ and Φ[xi] = αi.
Lemma 3.2. For every ring R there is a unique ring homomorphism Z → R.
Lemma 3.3. If R is a ring and a ∈ R then {ra : r ∈ R} = (a) is an ideal.
Theorem 3.4. A ring R is a field if and only if it has exactly two ideals.
Corollary 3.5. Let F be a field and R a non-zero ring. Then every homo- morphism ϕ : F → R is injective.
Lemma 3.6. Every ideal in Z is principle.
Theorem 3.7. Let g(x) be a monic polynomial in R[x] and let α be an element of R such that g(α) = 0.Then x − α divides g(x).
Theorem 3.8. If F is a field then every ideal of F [x] is principle.
Corollary 3.9. Let F be a field and let f, g ∈ F [x] which are both non-zero. Then there is a unique monic d(x) ∈ F [x] called the greatest common divisor of f, g such that
(a) d generates the ideal (f, g) of F [x] generated by f, g.
(b) d divides f and g
(c) If h is any divisor of f and g then h divides d.
(d) There are p, q ∈ F [x] such that d = pf + qg
