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Abstract Algebra (Dummit and Foote) Definitions and Theorems Chapter 3
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Kernel of a Homomorphism
The kernel is the set of elements of G mapping to the identity in H under a homomorphism. (If φ is a homomorphism φ: G -> H, the kernel of φ is the set {g∈G | φ(g) = 1}.)
Quotient Group
Let φ: G -> H be a homomorphism with kernel K. The quotient group (G/K) is the group whose elements are the fibers of φ with group operation defined above: namely if X is the fiber above a and Y is the fiber above b then the product of X with Y is defined to be the fiber above the product ab.
Proposition 3.2 (Multiplication of Fiber Representatives)
Let φ: G -> H be a homomorphism of groups with kernel K. Let X∈G/K be the fiber above a (i.e. X = φ^-1 (a).) Then:
(1) For any u∈X, X = {uk | k∈K}
(2) For any u∈X, X = {ku | k∈K}
Coset
For any N ≤ G and any g∈G, let gN = {gn | n∈N} and Ng = {ng | n∈N} called respectively a left coset and a right coset of N in G. Any element of a coset is called a representative for the coset.
Theorem 3.3 (Coset Representative Independence)
Let G be a group and let K be the kernel of some homomorphism from G to another group. Then the set whose elements are the left cosets of K in G with operation defined by uK ∘ vK = (uv)K forms a group, G/K.
In particular, this operation is well defined in the sense that if u1 is any element in uK and v1 is any element in vK, then u1v1 ∈ uvK, i.e. u1v1K = uvK so that the multiplication does not depend on the choice of representatives for the cosets.
Proposition 3.4 (Cosets Partition G)
Let N be any subgroup of the group G. The set of left cosets of N in G form a partition of G.
Furthermore, for all u,v ∈ G, uN = vN if and only if (v^-1)(u)∈N and in particular, uN = vN if and only if u and v are representatives of the same coset.
Proposition 3.5 (Conjugation and Coset Group)
Let G be a group and let N be a subgroup of G.
(1) The operation of the set of left cosets of N in G is described by uN · vN = (uv)N is well defined if and only if gng^-1 ∈ N for all g ∈ G and all n ∈ N.
(2) If the above operation is well defined, then it makes the set of left cosets of N in G into a group. In particular the identity of this group is the coset 1N and the inverse of gN is the coset (g^-1)N i.e. (gN)^-1 = (g^-1)N.
Normal
The element gng^-1 is called the conjugate of n∈N by g.
The set gNg^-1 = {gng^-1 | n∈N} is called the conjugate of N by g.
The element g is said to normalize N if gNg^-1 = N.
A subgroup N of a group G is called normal if every element of G normalizes N, i.e. if gNg^-1 = N for all g∈G.
If N is a normal subgroup of G we shall write N ◅ G.
Theorem 3.6 (Five Normality Equivalences)
Let N be a subgroup of the group G. The following are equivalent:
(1) N ◅ G
(2) The normalizer in G of N (N_G(N)) = G [the normalizer in G of N is G]
(3) gN = Ng for all g∈G. [cosets commutative]
(4) The operation of left cosets of N in G described in Proposition 5 makes the set of left cosets into a group. [left cosets group]
(5) gNg^-1 is contained in N for all g∈G. [conjugate of N with G always in N]
Proposition 3.7 (Normal Subgroups are Kernels)
A subgroup N of the group G is normal if and only if it is the kernel of some homomorphism.
Natural Projection
Let N ◅ G. The homomorphism π: G -> G/N defined by π(g) = gN is called the natural projection (homomorphism) of G onto G/N.
If A is a subgroup of G/N, the complete preimage of A in G is the preimage of A under the natural projection homomorphism.
3.1 Examples (Normal Subgroups)
Let G be a group.
(1) The subgroups 1 and G are always normal in G. G/1 is isomorphic to G and G/G is isomorphic to 1.
(2) If G is an abelian group, any subgroup N of G is normal.
(3) If N is a subgroup of Z(G), then N ◅ G.
Lagrange's Theorem
If G is a finite group and H is a subgroup of G, then the order of H divides the order of G and the number of left cosets of H in G equals |G|/|H|.
Index of H in G
If G is a group (possibly infinite) and H is a subgroup of G, the number of left cosets of H in G is called the index of H in G and is denoted by |G : H|.
Corollary 3.9 (Order of X divides order of G)
If G is a finite group and x∈G, then the order of x divides the order of G.
In particular, x^|G| = 1 for all x in G.
Corollary 3.10 (Primality and Cyclicality)
If G is a group of prime order p, then G is cyclic, hence G is isomorphic to Z/pZ.
3.2 Examples (Normality and Index)
(1) Let H = <(1 2 3)> be a subgroup of S3 and let G = S3. H ◅ S3.
(2) Let G be any group containing a subgroup H of index 2. H ◅ G.
(3) The property "is a normal subgroup of" is not transitive. For example, <s> ◅ <s, r^2> ◅ D8. (each subgroup is of index 2 in the next), however, <s> is not normal in D8 because rsr^-1 = sr^2 ∉ <s>.
3.2 More Examples (Searching for Normal Subgroups Harder)
(1) Let H = <(1 2)> be a subgroup of S3. Since H is of prime index 3 in S3, by Lagrange's Theorem, the only possibilities for N_S3(H) are H or S3.
(2) In D8 the only subgroup of order 2 which is normal is the center <r^2>.
Cauchy's Theorem
If G is a finite group and p is a prime dividing |G|, then G has an element of order p.
Sylow's Theorem
If G is a finite group of order (p^α)m, where p is a prime and p does not divide m, then G has a subgroup of order p^α.
HK
Let H and K be subgroups of a group and define HK = {hk | h∈H, k∈K}
Proposition 3.13 (Order Fraction)
If H and K are finite subgroups of a group, then |HK| = (|H||K|)/(|H intersect K|)
Proposition 3.14 (Subgroup Commutativity Requirement)
If H and K are subgroups of a group, HK is a subgroup if and only if HK = KH.
Corollary 3.15 (Normalizer and Subgroups)
If H and K are subgroups of G and H is a subgroup of N_G(K), then HK is a subgroup of G.
In particular, if K◅G, then HK is a subgroup of G for any H that is a subgroup of G.
Normalizes and Centralizers
If A is any subset of N_G(K) (or C_G(K)), we shall say A normalizes K (or centralizes K).
With this terminology, Corollary 15 states that HK is a subgroup if H normalizes K (similarly, HK is a subgroup if K normalizes H)
(The) First Isomorphism Theorem
If φ: G-> H is a homomorphism of groups, then kerφ ◅ G and G/kerφ is isomorphic to φ(G).
Corollary 3.17 (Homomorphism Facts)
Let φ: G -> H be a homomorphism of groups.
(1) φ is injective if and only if kerφ = 1
(2) |G : kerφ| = |φ(G)|
(The) Second Isomorphism Theorem
Let G be a group, let A and B be subgroups of G and assume A is a subgroup of N_G(B).
Then AB is a subgroup of G, B ◅ AB, A∩B ◅ A, and AB/B is isomorphic to A/A∩B.
(The) Third Isomorphism Theorem
Let G be a group and let H and K be normal subgroups of G with H being a subgroup of K.
Then K/H ◅ G/H and (G/H)/(K/H) is isomorphic to G/K.
(The) Fourth Isomorphism Theorem
Let G be a group and let N be a normal subgroup of G.
Then there is a bijection from the set of subgroups A of G which contain N onto the set of subgroups A' = A/N of G/N.
In particular, every subgroup of G' is of the form A/N for some subgroup A of G containing N (namely, its preimage in G under the natural projection homomorphism from G to G/N).
[Check the book for the rest of the definition]
Proposition 3.1 (Five Homomorphism Facts)
Let G and H be groups and let phi: G -> H be a homomorphism. Then:
1. phi(1_G) = 1_H where 1_G and 1_H are the identities of G and H respectively
2. phi(g^-1) = phi(g^)-1 for all g in G.
3. phi(g^n) = phi(g)^n for all integer n.
4. ker(phi) is a subgroup of G.
5. the image of G under phi, i.e., im(phi), is a subgroup of H.
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