In these “Group TheoryNotes PDF”, you will study an in-depth understanding of one of the most important branch of the abstract algebra with applications to practical real-world problems. Classification of all finite abelian groups (up to isomorphism) can be done.
The topics we will cover will be taken from the following list:
Automorphisms and Properties: Automorphism, inner automorphism, Automorphism groups, Automorphism groups of finite and infinite cyclic groups, Characteristic subgroups, Commutator subgroup and its properties; Applications of factor groups to automorphism groups.
External and Internal Direct Products of Groups: External direct products of groups and its properties, The group of units modulo n as an external direct product, Applications to data security and electric circuits; Internal direct products, Classification of groups of order KC , where K is a prime; Fundamental theorem of finite abelian groups and its isomorphism classes.
Group Action: Group actions and permutation representations; Stabilizers and kernels of group actions; Groups acting on themselves by left multiplication and consequences; Conjugacy in Sn.
Sylow Theorems and Applications: Conjugacy classes, Class equation, K-groups, Sylow theorems and consequences, Applications of Sylow theorems; Finite simple groups, Nonsimplicity tests; Generalized Cayley’s theorem, Index theorem, Embedding theorem and applications. Simplicity of A5.
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