Part A - TranquilEducationย ยท Part A Define Algebraic structure. The operations and relations on...
Transcript of Part A - TranquilEducationย ยท Part A Define Algebraic structure. The operations and relations on...
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Unit 4 Groups Part A
Define Algebraic structure. The operations and relations on the set ๐ define a structure on the elements of ๐, an algebraic system is called an algebraic structure. Define Semi-group Let S be a nonempty set and ๐ be a binary operation on S. The algebraic system (๐, . ) is called a semigroup if the operation . is associative. In other words (๐, . ) is a semigroup if for any ๐ฅ, ๐ฆ, ๐ง โ ๐,
๐ฅ. ๐ฆ . ๐ง = ๐ฅ. (๐ฆ. ๐ง). Define Monoid A semigroup (M,.) with an identity element with respect to the operation o is called a monoid. In other words, an algebraic system (M,.) is called a monoid if for any ๐ฅ, ๐ฆ, ๐ง โ ๐, ๐ฅ. ๐ฆ . ๐ง = ๐ฅ. (๐ฆ. ๐ง) and there exists an element ๐ โ ๐ such that for any ๐ฅ โ ๐, ๐. ๐ฅ = ๐ฅ. ๐ = ๐ฅ Define semigroup homomorphism. Let (๐,โ) and (๐, โ ) be any two semigroups. A mapping ๐: ๐ โ ๐ such that for any two elements ๐, ๐ โ ๐, ๐(๐ โ ๐) = ๐(๐) โ ๐(๐) is called a semigroup homomorphism. Define direct product Let (๐,โ) and (๐, โ) be two semigroups. The direct product of (๐,โ) and (๐, โ) is the algebraic system (๐ ร ๐, . ) in which the operation . on ๐ ร ๐ is defined by ๐ 1, ๐ก1 . ๐ 2, ๐ก2 = (๐ 1 โ ๐ 2 , ๐ก1โ ๐ก2) for any ๐ 1, ๐ก1 ๐๐๐ ๐ 2, ๐ก2 โ ๐ ร ๐. Show that the set ๐ต of natural numbers is a semigroup under the operation ๐ โ ๐ = ๐๐๐{๐, ๐}. Is it monoid? Given the operation ๐ฅ โ ๐ฆ = ๐๐๐ฅ{๐ฅ, ๐ฆ} for any ๐ฅ, ๐ฆ โ ๐. Clearly (๐,โ) is closed because ๐ฅ โ ๐ฆ = ๐๐๐ฅ{๐ฅ, ๐ฆ} โ ๐ and โ is associative as
(๐ฅ โ ๐ฆ) โ ๐ง = ๐๐๐ฅ{๐ฅ โ ๐ฆ, ๐ง} = ๐๐๐ฅ{๐๐๐ฅ{๐ฅ, ๐ฆ}, ๐ง}
= ๐๐๐ฅ{๐ฅ, ๐ฆ, ๐ง} = ๐๐๐ฅ{๐ฅ, ๐๐๐ฅ{๐ฆ, ๐ง}}
= ๐๐๐ฅ{๐ฅ, {๐ฆ โ ๐ง}} = ๐ฅ โ (๐ฆ โ ๐ง)
Therefore, ๐,โ is a semi-group. The identity ๐ of ๐,โ must satisfy the property that ๐ฅ โ ๐ = ๐ โ ๐ฅ = ๐ฅ. But ๐ฅ โ ๐ = ๐ โ ๐ฅ = ๐๐๐ฅ ๐ฅ, ๐ = ๐๐๐ฅ{๐, ๐ฅ} = ๐ฅ. Prove that โ A semi-group homomorphism preserves the property of associativity. ๐ฟ๐๐ก ๐, ๐, ๐ โ ๐,
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๐( ๐ โ ๐ โ ๐ = ๐ ๐ โ ๐ . ๐(๐)
= [ ๐ ๐ . ๐ ๐ . ๐(๐)] โฆ (1) ๐ ๐ โ ๐ โ ๐ = ๐ ๐ . ๐(๐ โ ๐)
= ๐ ๐ . [๐ ๐ . ๐(๐)] โฆ (2) ๐ต๐ข๐ก ๐๐ ๐, (๐ โ ๐) โ ๐ = ๐ โ (๐ โ ๐), โ ๐, ๐, ๐ โ ๐
โด ๐[(๐ โ ๐) โ ๐] = ๐[๐ โ (๐ โ ๐)] โ ๐ ๐ . ๐ ๐ .๐ ๐ = ๐ ๐ . [๐ ๐ . ๐(๐)]
โด The property of associativity is preserved. Prove that a semi group homomorphism preserves idem potency. Let ๐ โ ๐ be an idempotent element. โด ๐ โ ๐ = ๐
๐(๐ โ ๐) = ๐ ๐ . ๐(๐) = ๐(๐) โด ๐(๐ โ ๐) = ๐(๐).
This shows that ๐(๐) is an idempotent element in T. The property of idem potency is preserved under semi group homomorphism. Prove that A semigroup homomorphism preserves commutativity. Let ๐, ๐ โ ๐ Assume that ๐ โ ๐ = ๐ โ ๐
๐(๐ โ ๐) = ๐(๐ โ ๐) ๐ ๐ . ๐ ๐ = ๐ ๐ . ๐(๐).
This means that the operation . is commutative in ๐ The semigroup homomorphism preserves commutativity. Define group. A non-empty set G, together with a binary operation * is said to be a group if it satisfies the following axioms. i) โa,b โG โ a*b โ G (Closure Property) ii) For any ๐, ๐, ๐ โ ๐บ, (๐ โ ๐) โ ๐ = ๐ โ (๐ โ ๐) (Associative property) iii) There exists an element ๐ ๐๐ ๐บ such that ๐ โ ๐ = ๐ โ ๐ = ๐, โ๐ โ ๐บ (Identity) iv) For all ๐ โ ๐บ there exists an element ๐ โ 1 โ ๐บ such that ๐ โ ๐โ1 = ๐โ1 โ ๐ = ๐ (Inverse Property) Define Abelian group A Group (๐บ,โ) is said to be abelian if ๐ โ ๐ = ๐ โ ๐ for all ๐, ๐ โ ๐บ Define Left coset of H in G Let (๐ป,โ) be a subgroup of (๐บ,โ). For any ๐ โ ๐บ , the set ๐๐ป defined by
๐๐ป = {๐ โ ๐ / ๐ โ ๐ป } is called the left coset of ๐ป in ๐บ determined by the element
๐ โ ๐บ.
The element ๐ is called the representative element of the left coset ๐๐ป.
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State Lagrangeโs theorem The order of a subgroup of a finite group divides the order of the group. Or If ๐บ is a finite group, then ๐(๐ป) \ ๐(๐บ) , for all sub-group ๐ป of ๐บ. If (๐ฎ,โ) is a finite group of order ๐, then for any ๐ โ ๐ฎ, we have ๐๐ = ๐ , where e is the identity of the group ๐ฎ. Let ๐(๐บ) = ๐ and Let ๐ โ ๐บ Then order of the subgroup < ๐ > is the order of the element ๐. If ๐(< ๐ >) = ๐, then ๐๐ = ๐ and by Lagrangeโs theorem , we get ๐\ ๐.Let ๐ = ๐๐ Then ๐๐ = ๐๐๐ = (๐๐ )๐ = ๐๐ = ๐. Let ๐ฎ = ๐, ๐, ๐๐, ๐๐ ๐๐๐๐๐ (๐๐ = ๐) be a group and ๐ฏ = {๐, ๐๐} is a subgroup of G under multiplication . Find all the cosets of H. Let us find the right cosets of ๐ป in ๐บ.
๐ป1 = {1, ๐2} = ๐ป ๐ป๐ = {๐, ๐3}
๐ป๐2 = {๐2 , ๐4} = {๐2 , 1} = ๐ป ๐๐๐ ๐ป๐3 = {๐3, ๐5} = {๐3 , ๐} = ๐ป๐
โด ๐ป. 1 = ๐ป = ๐ป๐2 = {1, ๐2} ๐๐๐ ๐ป๐ = ๐ป๐3 = {๐, ๐3} are distinct right cosets of ๐ป in ๐บ. Similarly, we can find the left cosets of ๐ป in ๐บ. Find the left cosets of {[๐], [๐]} in the group (๐๐, +๐). Let ๐4 = {[0], [1], [2], [3]} be a group and ๐ป = { [0], [2]} be a sub-group of ๐4 under +4. The left cosets of ๐ป are
[0] + ๐ป = {[0], [2]} [1] + ๐ป = {[1], [3]}
[2] + ๐ป = {[2], [4]} = {[2], [0]} = {[0], [2]} = ๐ป [3] + ๐ป = {[3], [5]} = {[3], [1]} = {[1], [3]} = [1] + ๐ป
[0] + ๐ป = [2] + ๐ป = ๐ป ๐๐๐ [1] + ๐ป = [3] + ๐ป are the two distinct left cosets of ๐ป in ๐4. Define subgroup Let (๐บ,โ) be a group and let ๐ป be a non-empty subset of ๐บ. Then ๐ป is said to be a subgroup of ๐บ if ๐ป itself is a group with respect to the operation โ . Define normal subgroup A subgroup ๐ป,โ of ๐บ,โ is called a normal sub-group if for any ๐ โ ๐บ, ๐๐ป = ๐ป๐. (i.e.) Left coset = Right coset Prove that every subgroup of an abelian group is normal subgroup. Let (๐บ,โ) be an abelian group and (๐,โ) be a subgroup of ๐บ. Let ๐ be any element in ๐บ and let ๐ โ ๐. Now ๐ โ ๐ โ ๐โ1 = (๐ โ ๐) โ ๐โ1 [Since G is abelian] = ๐ โ ๐ = ๐ โ ๐
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โด โ ๐ โ ๐บ ๐๐๐ ๐ โ ๐, ๐ โ ๐ โ ๐โ1 โ ๐ โด (๐,โ) is a normal subgroup. Define direct product on groups Let (๐บ,โ) and (๐ป, โ) be two groups. The direct product of these two groups is the algebraic structure ( ๐บ ร ๐ป , . ) in which the binary operation . on ๐บ ร ๐ป is given by ๐1, ๐1 . (๐2, ๐2) = (๐1 โ ๐2, ๐1โ ๐2) for any ๐1, ๐1 , (๐2, ๐2) โ ๐บ ร ๐ป. If S denotes the set of positive integers โค 100, for any ๐, ๐ โ ๐บ, define ๐ โ ๐ = ๐๐๐{๐, ๐}. Verify whether (๐บ,โ) is a monoid assuming that โ is associative. The identity element is ๐ = 100 exists. Since for ๐ฅ โ ๐, ๐๐๐ (๐ฅ, 100) = ๐ฅ โ ๐ฅ โ 100 = ๐ฅ , โ๐ฅ โ ๐ If ๐ฏ is a subgroup of the group ๐ฎ, among the right cosets of ๐ฏ in ๐ฎ. Prove that there is only one subgroup viz., ๐ฏ. Let ๐ป๐ be a right coset of ๐ป in ๐บ where ๐ โ ๐บ. If ๐ป๐ is a subgroup of ๐บ then ๐ โ ๐ป๐, where ๐ is the identity element in ๐บ. ๐ป๐ is an equivalence class containing ๐ with respect to an equivalence relation.
๐ โ ๐ป๐ โ ๐ป. ๐ = ๐ป๐. But ๐ป๐ = ๐ป โด ๐ป๐ = ๐ป. ๐๐๐๐ ๐ ๐๐๐ค๐ ๐ป ๐๐ ๐๐๐๐ฆ ๐ ๐ข๐๐๐๐๐ข๐.
Give an example of sub semi-group For the semi group (๐, +), where ๐ is the set of natural number, the set ๐ธ of all even non-negative integers (๐ธ, +) is a sub semi-group of (๐, +). Find the subgroup of order two of the group (๐๐, +๐) ๐ป = { [0], [4]} is a subgroup of order two of the group ๐บ = (๐8, +8) .
+๐ [๐] [๐] [๐] [0] [4]
[๐] [4] [0]
Define Ring An algebraic system (๐, +, . ) is called a ring if the binary operations + and . on ๐ satisfy the following three properities. i)(๐, +) is an abelian group ii)(๐, . ) is a semigroup iii) The operation . is distributive over + , i.e. , for any ๐, ๐, ๐ โ ๐ ,
๐. (๐ + ๐) = ๐. ๐ + ๐. ๐ ๐๐๐ (๐ + ๐). ๐ = ๐. ๐ + ๐. ๐ Define Subring
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A commutative ring (๐, +, . ) is a ring is called a subring if (๐ , +, . ) is itself with the operations + and . restricted to ๐ . Define Ring homomorphism Let (๐ , +, . ) ๐๐๐ (๐,โ,โ) be rings. A mapping g:RโS is called a ring homomorphism from (๐ , +, . ) to (๐,โ,โ) if for any ๐, ๐ โ ๐ .
๐(๐ + ๐) = ๐(๐) โ ๐(๐) ๐๐๐ ๐(๐. ๐) = ๐(๐) โ ๐(๐) If (๐น, +, . ) be a ring then prove that ๐ . ๐ = ๐ for every ๐ โ ๐น Proof: Let ๐ โ ๐ then ๐. 0 = ๐. (0 + 0) = ๐. 0 + ๐. 0 [ by Distributive Law ] ๐. 0 = 0 [ Cancellation Law ] Give an example of an ring with zero-divisors. The ring ( ๐10 , +10 , .10 ) is not an integral domain. Since 5 .10 2 = 0 , ( 5 โ 0, 2 โ 0 ๐๐ ๐10)
Define Field. The commutative ring (๐ , +,ร) with unity is said to be a Field if it has inverse element under the binary operation ร. ๐โ1 ร ๐ = ๐ ร ๐โ1 = 1, โ๐ โ ๐ .
Part B State and Prove Lagrangeโs theorem for finite groups. Statement: The order of a subgroup of a finite group is a divisor of the order of the group. Proof: Let ๐๐ป and ๐๐ป be two left cosets of the subgroup {๐ป,โ} in the group {๐บ,โ}. Let the two cosets ๐๐ป and ๐๐ป be not disjoint. Then let ๐ be an element common to ๐๐ป and ๐๐ป i.e., ๐ โ ๐๐ป โฉ ๐๐ป
โต ๐ โ ๐๐ป, ๐ = ๐ โ ๐1, ๐๐๐ ๐ ๐๐๐ ๐1 โ ๐ป โฆ (1) โต ๐ โ ๐๐ป, ๐ = ๐ โ ๐2, ๐๐๐ ๐ ๐๐๐ ๐2 โ ๐ป โฆ (2)
From (1) and (2), we have ๐ โ ๐1 = ๐ โ ๐2
๐ = ๐ โ ๐2 โ ๐1โ1 โฆ (3)
Let ๐ฅ be an element in ๐๐ป ๐ฅ = ๐ โ ๐3, ๐๐๐ ๐ ๐๐๐ ๐3 โ ๐ป
= ๐ โ ๐2 โ ๐1โ1 โ ๐3, ๐ข๐ ๐๐๐ (3)
Since H is a subgroup, ๐2 โ ๐1โ1 โ ๐3 โ ๐ป
Hence, (3) means ๐ฅ โ ๐๐ป Thus, any element in ๐๐ป is also an element in ๐๐ป. โด ๐๐ป โ ๐๐ป Similarly, we can prove that ๐๐ป โ ๐๐ป Hence ๐๐ป = ๐๐ป Thus, if ๐๐ป and ๐๐ป are disjoint, they are identical.
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The two cosets ๐๐ป and ๐๐ป are disjoint or identical. โฆ(4) Now every element ๐ โ ๐บ belongs to one and only one left coset of ๐ป in ๐บ, For, ๐ = ๐๐ โ ๐๐ป, ๐ ๐๐๐๐ ๐ โ ๐ป โ ๐ โ ๐๐ป
๐ โ ๐๐ป, since ๐๐ป and ๐๐ป are disjoint i.e., ๐ belongs to one and only left coset of ๐ป in ๐บ i.e., ๐๐ป โฆ (5) From (4) and (5), we see that the set of left cosets of ๐ป in ๐บ form the partition of ๐บ. Now let the order of ๐ป be ๐. Let ๐ป = ๐1, ๐2, โฆ , ๐๐ , ๐ค๐๐๐๐ ๐๐ โฒ๐ are distinct Then ๐๐ป = ๐๐1, ๐๐2, โฆ , ๐๐๐ The elements of ๐๐ป are also distinct, for, ๐๐๐ = ๐๐๐ โ ๐๐ = ๐๐ , which is not
true. Thus ๐ป and ๐๐ป have the same number of elements, namely ๐. In fact every coset of ๐ป in ๐บ has exactly ๐ elements. Now let the order of the group {๐บ,โ} be ๐, i.e., there are ๐ elements in ๐บ Let the number of distinct left cosets of ๐ป in ๐บ be ๐. โด The total number of elements of all the left cosets = ๐๐ = the total number of elements of ๐บ. i.e., ๐ = ๐๐ i.e., ๐, the order of ๐ป is adivisor of ๐, the order of ๐บ. Find all non-trivial subgroups of (๐๐ , +๐) Solution: ๐6 , +6 , ๐ = 0 ๐ข๐๐๐๐ ๐๐๐๐๐๐ฆ ๐๐๐๐๐๐ก๐๐๐ +6 are trivial subgroups
+๐ [0] [1] [2] [3] [4] [5]
[0] [0] [1] [2] [3] [4] [5]
[1] [1] [2] [3] [4] [5] [0]
[2] [2] [3] [4] [5] [0] [1]
[3] [3] [4] [5] [0] [1] [2]
[4] [4] [5] [0] [1] [2] [3]
[5] [5] [0] [1] [2] [3] [4]
๐1 = 0 , 2 , [4]
+๐ [0] [2] [4]
[0] [0] [2] [4]
[2] [2] [4] [0]
[4] [4] [0] [2]
From the above cayleyโs table, All the elements are closed under the binary operation +๐ Associativity is also true under the binary operation +๐ [0] is the identity element. Inverse element of [2] is [4] and vise versa
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Hence ๐1 = 0 , 2 , [4] is a subgroup of (๐6 , +6) ๐2 = 0 , 3
+๐ [0] [3]
[0] [0] [3]
[3] [3] [0]
From the above cayleyโs table, All the elements are closed under the binary operation +๐ Associativity is also true under the binary operation +๐ [0] is the identity element. Inverse element of [3] is itself. Hence ๐2 = 0 , 3 is a subgroup of (๐6 , +6) Therefore ๐1 = 0 , 2 , [4] and ๐2 = 0 , 3 are non trivial subgroups of
(๐6 , +6) Show that the mapping ๐: (๐บ, +) โ (๐ป,โ) defined by ๐(๐) = ๐๐, where ๐บ is the set
of all rational numbers under addition operation + and ๐ป is the set of non-zero real numbers under multiplication operation โ is a homomorphism but not isomorphism. Solution: For any ๐, ๐ โ ๐,
๐ ๐ + ๐ = 3๐+๐ = 3๐ โ 3๐ = ๐ ๐ โ ๐(๐) โด ๐ is a homomorphism. To prove ๐ is one to one: For any ๐, ๐ โ ๐, Let ๐ ๐ = ๐ ๐ โ 3๐ = 3๐ โ๐ = ๐
โด ๐ is one to one To prove ๐ is onto:
๐ = 3๐ โ log๐ = log 3๐ โ log๐ = a log 3 โ๐ =log๐
log 3
โด ๐ = ๐ log๐
log 3 , โ๐ โ ๐
โด โ๐ โ ๐, there is a pre-image log ๐
log 3โ ๐
โต log 3 is irratioinal โlog ๐
log 3 is irratioinal
โด ๐ is not onto. โด ๐ is not an isomorphism.
The intersection of any two subgroups of a group G is again a subgroup of G. โProve. Proof: Let ๐ป1 and ๐ป2 be two normal subgroups of a group (๐บ,โ). Then ๐ป1 ๐๐๐ ๐ป2 are subgroups.
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๐ โ ๐ป1 ๐๐๐ ๐ โ ๐ป2 โ ๐ โ ๐ป1 โฉ ๐ป2 . Since ๐ is the identity element of G and it is
unique. โด ๐ป1 โฉ ๐ป2 ๐๐ ๐๐๐ ๐๐๐๐ก๐ฆ.
โ๐, ๐ โ ๐ป1 โฉ ๐ป2 โ๐, ๐ โ ๐ป1 ๐๐๐ ๐, ๐ โ ๐ป2 โ๐ โ ๐โ1 โ ๐ป1 ๐๐๐ ๐ โ ๐โ1 โ ๐ป2
Since ๐ป1 ๐๐๐ ๐ป2 are subgroups. โ ๐ โ ๐โ1 โ ๐ป1 โฉ ๐ป2
โด ๐ป1 โฉ ๐ป2 ๐๐ ๐ ๐ ๐ข๐๐๐๐๐ข๐ Show that monoid homomorphism preserves the property of invertibility. Solution: If ๐,โ, ๐ ๐๐๐ ๐,โ , ๐ โฒ ๐๐ ๐๐๐ฆ ๐ก๐ค๐ ๐๐๐๐๐๐๐ , ๐ค๐๐๐๐ ๐ ๐๐๐ ๐ โฒ๐๐๐ ๐๐๐๐๐ก๐๐ก๐ฆ elements of ๐ and ๐ with respect to the operations โ and . respectively, then a mapping ๐: ๐ โ ๐ such that, for any two elements ๐, ๐ โ ๐,
๐ ๐ โ ๐ = ๐ ๐ . ๐ ๐ ๐๐๐ ๐ ๐ = ๐โฒ is called monoid homomorphism. Let ๐โ1 โ ๐ be the inverse of ๐ โ ๐ Then ๐ ๐ โ ๐โ1 = ๐ ๐ = ๐ โฒ ๐๐ฆ ๐๐๐๐๐๐๐ก๐๐๐. Also ๐ ๐ โ ๐โ1 = ๐ ๐ . ๐ ๐โ1 ๐๐ฆ ๐๐๐๐๐๐๐ก๐๐๐
๐ ๐ . ๐ ๐โ1 = ๐โฒ Hence the inverse of ๐ ๐ = ๐ ๐โ1 = ๐(๐) โ1 โด Monoid homomorphism preserves the property of invertibility. Prove that the intersection of two normal subgroup of a group will be a normal subgroup. Solution: Let ๐ป1 and ๐ป2 be two normal subgroups of a group (๐บ,โ). Then ๐ป1 ๐๐๐ ๐ป2 are subgroups. Since ๐ โ ๐ป1 ๐๐๐ ๐ โ ๐ป2 โ ๐ โ ๐ป1 โฉ ๐ป2
โด ๐ป1 โฉ ๐ป2 ๐๐ ๐๐๐ ๐๐๐๐ก๐ฆ. โ๐, ๐ โ ๐ป1 โฉ ๐ป2 โ ๐, ๐ โ ๐ป1 ๐๐๐ ๐, ๐ โ ๐ป2โ ๐ โ ๐โ1 โ ๐ป1 ๐๐๐ ๐ โ ๐โ1 โ ๐ป2
Since ๐ป1 ๐๐๐ ๐ป2 are subgroups. โ ๐ โ ๐โ1 โ ๐ป1 โฉ ๐ป2
โด ๐ป1 โฉ ๐ป2 ๐๐ ๐ ๐ ๐ข๐๐๐๐๐ข๐ โ๐ โ ๐บ, โ๐ โ ๐ป1 โฉ ๐ป2 โ ๐ โ ๐ป1 ๐๐๐ ๐ โ ๐ป2 , โ ๐โ1 โ ๐ โ ๐ โ ๐ป1 ๐๐๐ ๐โ1 โ ๐ โ ๐ โ ๐ป2 Since ๐ป1 ๐๐๐ ๐ป2 are normal subgroups.
โ ๐โ1 โ ๐ โ ๐ โ ๐ป1 โฉ ๐ป2 โด ๐ป1 โฉ ๐ป2 ๐๐ ๐ ๐๐๐๐๐๐ ๐ ๐ข๐๐๐๐๐ข๐
Let ๐บ be a non-empty set and ๐ท(๐บ) denote the power set of ๐บ. Verify that (๐ท(๐บ),โฉ) is a group. Solution: โต ๐(๐) denote the power set of ๐ โ๐ด, ๐ต โ ๐(๐) โ ๐ด โฉ ๐ต โ ๐(๐)
โด ๐(๐) is closed.
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โ๐ด, ๐ต, ๐ถ โ ๐(๐) โ ๐ด โฉ ๐ต โฉ ๐ถ = ๐ด โฉ ๐ต โฉ ๐ถ
โด ๐(๐) is associative โ๐ด โ ๐(๐) , we have ๐ด โฉ ๐ = ๐ด = ๐ โฉ ๐ด โด ๐ โ ๐(๐) be the identity element. โ๐ด โ ๐ ๐ , there exists some ๐ต โ ๐ ๐ such that
๐ด โฉ ๐ต โ ๐ โด Inverse does not exists for any subset except ๐ (๐(๐),โฉ) is not a group but it is a monoid. Let (๐ฎ,โ) and (๐ฏ, โ) be groups and ๐: ๐ฎ โ ๐ฏ be a homomorphism. Then prove that kernel of ๐ is a normal sub-group of ๐ฎ. Solution: Let ๐พ = ๐๐๐ ๐ = {๐ ๐ = ๐ โฒ \๐ โ ๐บ, ๐ โฒ โ ๐ป} To prove ๐พ is a subgroup of ๐บ: We know that ๐ ๐ = ๐ โฒ โ ๐ โ ๐พ โด ๐พ is a non-empty subset of ๐บ. By the definition of homomorphism ๐ ๐ โ ๐ = ๐ ๐ โ ๐ ๐ , โ๐, ๐ โ ๐บ Let ๐, ๐ โ ๐พ โ ๐ ๐ = ๐ โฒ ๐๐๐ ๐ ๐ = ๐ โฒ
Now ๐ ๐ โ ๐โ1 = ๐ ๐ โ ๐ ๐โ1 = ๐ ๐ โ ๐ ๐ โ1
= ๐ โฒโ ๐ โฒ โ1
= ๐ โฒโ ๐ โฒ = ๐โฒ โด ๐ โ ๐โ1 โ ๐พ
โด ๐พ is a subgroup of ๐บ To prove ๐พ is a normal subgroup of ๐บ: For any ๐ โ ๐บ ๐๐๐ ๐ โ ๐พ,
๐ ๐โ1 โ ๐ โ ๐ = ๐ ๐โ1 โ ๐ ๐ โ๐ ๐ = ๐ ๐โ1 โ ๐ ๐ โ ๐ ๐ = ๐ ๐โ1 โ ๐ โฒโ ๐ ๐ = ๐ ๐โ1 โ ๐ ๐ = ๐ ๐โ1 โ ๐ = ๐ ๐ = ๐โฒ
๐โ1 โ ๐ โ ๐ โ ๐พ โด ๐พ is a normal subgroup of ๐บ State and Prove Fundamental theorem of homomorphism. Statement: Let ๐ be a homomorphism from a group (๐บ,โ) to a group (๐ป, โ), and let ๐พ be the kernel of ๐ and ๐ปโฒ โ ๐ป be the image set of ๐ in ๐ป. Then ๐บ/๐พ is isomorphic to ๐ปโฒ. Proof: Since ๐พ is the kernel of homomorphism, it must be a normal subgroup of G. Also we can define a mapping ๐: (๐บ,โ) โ (๐บ/๐พ,โ) where โ is defined as ๐ โ ๐ ๐ป = ๐๐ป โ๐๐ป, โ๐, ๐ โ ๐บ โฆ (1) i.e., ๐ ๐ = ๐๐พ ๐๐๐ ๐๐๐ฆ ๐ โ ๐บ โฆ (2) Let us define a mapping ๐: ๐บ/๐พ โ ๐ปโฒ such that ๐ ๐๐พ = ๐ ๐ โฆ (3) To prove that ๐ is well defined: For any ๐, ๐ โ ๐บ,
โด ๐๐พ = ๐๐พ ๐ โ ๐โ1 โ ๐พ โ ๐ โ ๐พ๐
๐ ๐ โ ๐โ1 = ๐โฒ ๐ ๐๐๐๐ ๐ ๐๐ ๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐๐๐๐๐๐ ๐ ๐๐๐๐ ๐บ ๐ก๐ ๐ป
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๐ ๐ โ ๐ ๐โ1 = ๐ โฒ ๐ ๐๐๐๐ ๐ ๐๐ ๐๐๐๐๐๐๐๐๐๐๐ ๐ ๐๐๐๐ ๐บ ๐ก๐ ๐ป
๐ ๐ โ ๐ ๐ โ1
= ๐ โฒ โต ๐ ๐ โ1
= ๐ ๐โ1
๐ ๐ โ ๐ ๐ โ1
โ ๐ ๐ = ๐ โฒโ ๐ ๐ ๐ ๐ โ ๐ โฒ = ๐ ๐ โ ๐ ๐ = ๐(๐)
๐ ๐๐พ = ๐ ๐๐พ ๐๐พ = ๐๐พ โ ๐ ๐๐พ = ๐ ๐๐พ
โด ๐ is well defined. To prove that ๐ is homomorphism:
๐ ๐๐พ โ ๐๐พ = ๐ ๐ โ ๐ ๐พ ๐๐๐๐(1)
= ๐ ๐ โ ๐ ๐๐๐๐(3) = ๐ ๐ โ ๐ ๐ ๐ ๐๐๐๐ ๐ ๐๐ ๐๐๐๐๐๐๐๐๐๐๐ ๐ ๐๐๐๐ ๐บ ๐ก๐ ๐ป
= ๐ ๐๐พ โ ๐ ๐๐พ ๐๐๐๐(3) โด ๐ is homomorphism To prove that ๐ is on to: The image set of the mapping ๐ is the same as the image set of the mapping ๐, so that ๐: ๐บ/๐พ โ ๐ปโฒ is on to. To prove that ๐ is one to one: For any ๐, ๐ โ ๐บ,
๐ ๐๐พ = ๐ ๐๐พ ๐ ๐ = ๐ ๐
๐ ๐ โ ๐ ๐ โ1
= ๐ ๐ โ ๐ ๐ โ1
๐ ๐ โ ๐ ๐โ1 = ๐โฒ ๐ ๐ โ1
= ๐ ๐โ1 & ๐ ๐ โ ๐ ๐ โ1
= ๐โฒ
๐ ๐ โ ๐โ1 = ๐โฒ ๐ ๐๐๐๐ ๐ ๐๐ ๐๐๐๐๐๐๐๐๐๐๐ ๐ ๐๐๐๐ ๐บ ๐ก๐ ๐ป ๐ โ ๐โ1 โ ๐พ โ ๐ โ ๐พ๐
โด ๐๐พ = ๐๐พ โด ๐ is one to one โด ๐: ๐บ/๐พ โ ๐ปโฒ is isomorphic. Show that every subgroup of a cyclic group is cyclic. Proof:
Let ๐บ be the cyclic group generated by the element ๐ and let ๐ป be a subgroup of ๐บ. If
๐ป = ๐บ ๐๐ {๐}, ๐ป is cyclic. If not the elements of ๐ป are non-zero integral powers of ๐,
since, if ๐๐๐ ๐ป , its inverse ๐โ๐๐ ๐ป.
Let ๐ be the least positive integer for which ๐๐๐ ๐ป
Now let ๐๐ be any arbitrary element of H. Let ๐ be the quotient and ๐ be the remainder
when ๐ is divided by ๐.
Then ๐ = ๐๐ + ๐, ๐ค๐๐๐๐ 0 โค ๐ < ๐
Since, ๐๐๐ ๐ป, ๐๐ ๐๐ ๐ป, ๐๐ฆ ๐๐๐๐ ๐ข๐๐ ๐๐๐๐๐๐๐ก๐ฆ
๐๐๐ ๐ ๐ป โ ๐๐๐ โ1๐ ๐ป, ๐๐ฆ ๐๐ฅ๐๐ ๐ก๐๐๐๐ ๐๐ ๐๐๐ฃ๐๐๐ ๐, ๐๐ ๐ป ๐๐ ๐ ๐ ๐ข๐๐๐๐๐ข๐
๐โ๐๐ ๐ ๐ป. Now since, ๐๐๐ ๐ป ๐๐๐ ๐โ๐๐ ๐ ๐ป โ๐๐โ๐๐ ๐ ๐ป โ๐๐๐ ๐ป
๐ = 0 โด ๐ = ๐๐
โด ๐๐ = ๐๐๐ = ๐๐ ๐
Thus, every element ๐๐๐ ๐ป is of the form ๐๐ ๐ .
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Hence H is a cyclic subgroup generated by ๐๐ . State and prove Cayleyโs theorem on permutation groups. Statement: Every group ๐บ is isomorphic to a subgroup of the group of permutation ๐๐ด for some set ๐ด. Proof: We know that ๐ โ ๐๐บ is the subgroup of permutation group ๐๐บ . We prove the result by choosing ๐ด to be ๐บ. In fact, we prove that the mapping ๐: ๐บ,โ โ ๐, ๐ given by ๐ ๐ = ๐๐ is an
isomorphism from ๐บ on to ๐. To prove ๐ is homomorphism: Let ๐, ๐ โ ๐บ, then
๐ ๐ โ ๐ = ๐๐โ๐ = ๐๐๐๐๐ = ๐ ๐ ๐๐ ๐ โด ๐ is homomorphism To prove ๐ is one to one:
๐ ๐ = ๐ ๐ ๐๐ = ๐๐ โ๐๐ ๐ = ๐๐ (๐)
๐ โ ๐ = ๐ โ ๐ ๐ = ๐
โด ๐ is one to one To prove ๐ is on to: โต ๐ ๐ = ๐๐ , For every image ๐๐ in ๐ there is a pre image ๐ in ๐บ. โด ๐ is on to. โด ๐ is isomorphism. Prove that every finite integral domain is a field. Proof:
Let ๐ท, +, . be a finite integral domain. Then ๐ท has a finite number of distinct elements,
say, ๐1, ๐2, โฆ , ๐๐ . Let ๐ โ 0 be an element of ๐ท.
Then the elements ๐. ๐1, ๐. ๐2 , โฆ , ๐. ๐๐๐ ๐ท, since ๐ท is closed under multiplication. The
elements ๐. ๐1, ๐. ๐2 , โฆ , ๐. ๐๐ are distinct, because if ๐. ๐๐ = ๐. ๐๐ , ๐ก๐๐๐
๐. ๐๐ โ ๐๐ = 0. But ๐ โ 0. Hence ๐๐ โ ๐๐ = 0, since ๐ท is an integral domain i.e.,
๐๐ = ๐๐ , which is not true, since ๐1, ๐2, โฆ , ๐๐ are distinct elements of ๐ท.
Hence the sets {๐. ๐1 , ๐. ๐2, โฆ , ๐. ๐๐ } and ๐1 , ๐2 , โฆ , ๐๐ are the same.
Since ๐๐ ๐ท is in both sets, let ๐. ๐๐ = ๐ for some ๐ โฆ (1)
Then ๐๐ is the unity of ๐ท, detailed as follows
Let ๐๐ = ๐. ๐๐ โฆ (2)
Now ๐๐ . ๐๐ = ๐๐ . ๐๐ , ๐๐ฆ ๐๐๐๐๐ข๐ก๐๐ก๐๐ฃ๐๐ก๐ฆ
= ๐๐ . ๐. ๐๐ by (2)
= ๐๐ . ๐ . ๐๐
= ๐. ๐๐ . ๐๐
= ๐. ๐๐ ๐๐ฆ (1)
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= ๐๐ ๐๐ฆ (2)
Since, ๐๐ is an arbitrary element of ๐ท
๐๐ is the unity of ๐ท
Let it be denoted by 1.
Since 1๐ ๐ท, there exist ๐ โ 0 and ๐๐๐ ๐ท such that ๐. ๐๐ = ๐๐ . ๐ = 1
๐ has an inverse.
Hence ๐ท, +, . is a field.