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GROUP THEORY (MATH 33300) COURSE NOTES. CONTENTS. Basics. Homomorphisms. Subgroups. Generators. Cyclic groups. Cosets and Lagrange’s Theorem. Normal subgroups and quotient groups. Isomorphism Theorems. Direct products. Group actions. Sylow’s Theorems. Applications of Sylow’s Theorems. Finitely generated abelian groups. The symmetric group.
number systems give prototypes for mathematical structures worthy of investigation. (R;+,·) and (Q;+,·) serve as examples of fields, (Z;+,·) is an example of a ring which is not a field.
Definition 1.1. A group (G, ∗) is a set G with a binary operation ∗ that has three requirements satisfied: 1. Associativity: a ∗ (b ∗ c) = (a ∗ b) ∗ c for all elements a, b, c ∈ G. 2. Identity: there is an element e ∈ G in which a ∗ e = e ∗ a = a for all elements of G. The identity for groups under multiplication is 1, under addition it is 0. 3.
This is well-defined: if gK = g0K then g0 = gk for some k 2 K and then f (g0) = f (gk) = f (g) f (k) = f (g) since k 2 K. It is a group homomorphism by definintion of the product structure on G=K. The image is the same as f by construction. As to the kernel, ̄f (gK) = eH iff f (g) = eH iff g 2 K iff gK = K = eG=K.
Let us start with some very simple examples. ACTION GROUPS. A nonempty collection of actions that can be performed one after another is called. a group if every action has a counteraction also included in this collection, and the result of performing any two of these actions in a row is also included in the collection. EXAMPLE 1. [5] “
every simple group has to be tested to see if it leads to new simple groups containing it in the centralizer of an involution. For example, when the baby monster was discovered, it had a double cover, which was a potential centralizer of an involution in a larger simple group, which turned out to be the monster. The monster happens to have no
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