MATH 436 Notes: Subgroups and Cosets
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MATH 436 Notes: Cyclic groups and Invariant Subgroups
Thus if x has infinite order then the elements in the set {x|k ∈ Z} are all distinct while if x has order o(x) ≥ 1 then x = e exactly when k is a multiple of o(x). Thus x = x whenever m ≡ n modulo o(x). By the First Isomorphism Theorem, < x >∼= Z/o(x)Z. Hence when x has finite order, | < x > | = o(x) and so in the case of a finite group, this order must divide the order of the group G by Lagran...
متن کاملIntroduction to Abstract Algebra ( Math 113 )
3 Groups 12 3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Subgroups, Cosets and Lagrange’s Theorem . . . . . . . . . . . . . . . . . . 14 3.3 Finitely Generated Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Permutation Groups and Group Actions . . . . . . . . . . . . . . . . . . . . 19 3.5 The Oribit-Stabiliser Theorem . ....
متن کاملNotes on Group Theory
2 Basic group theory 2 2.1 The definition of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Group homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.4 Cosets and quotient spaces . . . . . . . . . . . . . . . . . . . . . . . . . . ....
متن کاملMATH 436 Notes: Sylow Theory
We are now ready to apply the theory of group actions we studied in the last section to study the general structure of finite groups. A key role is played by the p-subgroups of a group. We will see that the Sylow theory will give us a way to study a group “a prime at a time”. First we record a very important special case of group actions: Theorem 1.1 (p-group Actions). Let p be a prime. If P is...
متن کاملThe lower and upper approximations in a group
In this paper, we generalize some propositions in [C.Z. Wang, D.G. Chen, A short note on some properties of rough groups, Comput. Math. Appl. 59(2010)431-436.] and we give some equivalent conditions for rough subgroups. The notion of minimal upper rough subgroups is introduced and a equivalent characterization is given, which implies the rough version of Lagrange′s Theorem. Keywords—Lower appro...
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