Cyclicgroup group thoery ppt (1)
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This document discusses cyclic groups and their applications. It defines what a group is by outlining the four properties a set and binary operation must satisfy to be considered a group: closure, associativity, identity, and inverses. It also defines subgroups as subsets of a group that themselves satisfy the group properties under the restriction of the operation to the subset. An example of the symmetric group S3 is given to illustrate group elements and their orders.
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3) It is shown that if an L-fuzzy subset A of an l-group G satisfies certain conditions, then A is an L-fuzzy normal sub l-group of G. Additionally, if A is an L-fuzzy normal sub l-
A common fixed point of integral type contraction in generalized metric spacess
This document presents a common fixed point theorem for two self-mappings S and T on a G-metric space X that satisfies a contractive condition of integral type. It begins with definitions related to G-metric spaces and contractive conditions. It then states Theorem 1.1, which proves that if S and T satisfy the given integral type contractive condition, along with other listed conditions, then S and T have a unique point of coincidence in X. If S and T are also weakly compatible, then they have a unique common fixed point. The proof of Theorem 1.1 is then provided.
Sets, functions and groups
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UTM JOURNAL
1. The document discusses the probability that an element of a metacyclic 2-group of positive type fixes a set. It defines metacyclic groups and the concept of a group acting on a set.
2. Previous work on the commutativity degree and the probability that a group element fixes a set is summarized. It was shown that this probability equals the number of conjugacy classes divided by the total number of sets.
3. The main results compute the probability that an element of a metacyclic 2-group of positive type fixes a set, by considering the action of the group on the set of commuting element pairs through conjugation.
cont.3rdppt.pptforabstractalegbrapañahah
V is not cyclic.
The Klein 4-group V = {e, a, b, c} where the binary operation is addition modulo 2.
It can be shown that <a> = {e, a} ≠ V, <b> = {e, b} ≠ V, and <c> = {e, c} ≠ V.
Since no element generates the entire group V, V is not cyclic.
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Cyclicgroup group thoery ppt (1)
- 2. APPLICATIONS OF GROUP THEORY: • Physical sciences • Biological Sciences • Study of Crystal Structure • Configuration of molecules • Structure of human Genes • And in Coding theory
- 3. What is a Group ? A group consistsofa setG together with a binary operation '∗'whichsatisfies (1) Closure Property: g ∗h ∈G for all g, h ∈G; (2) Associative Property : g ∗(h ∗k) =(g ∗h)∗k for all g, h, k ∈G; (3) Existence ofIdentity:there isan element 'e' in G which satisfies g ∗e =e ∗g =g for all g ∈ G ; we call 'e' the identity of G; (4) Existence of Inverse:for each g ∈ G there is an element g- 1 ∈G satisfying g ∗g-1 =g-1 ∗g = e ; we call g-1 the inverse ofg.
- 4. Subgroup: – In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is usually denoted H ≤ G, read as "H is a subgroup of G".
- 6. Examples:
- 7. Examples
- 8. EXAMPLE: • e s t u v w e e s t u v w s s e v w t u t t u e s w v u u t w v e s v v w s e u t w w v u t s e This group has six elements, so ord(S3) = 6. By definition, the order of the identity, e, is one, since e 1 = e. Each of s, t, and w squares to e, so these group elements have order two: |s| = |t| = |w| = 2. Finally, u and v have order 3, since u3 = vu = e, and v3 = uv = e.
- 13. An y
- 14. THE END