Abstract Algebra: An Introduction (3rd Edition)

Chapter 7

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, , , , ,.    ; According to the theorem, every ... more

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(a). (b). (c). (d). (e). ; (a).  Since the order ... more

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; Since , if is a element of then additive ... more

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; According to the definition . So, by using this... more

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; Operation table for after direct calculation ... more

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Group will be abelian if inverse exist for each ... more

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; All the operations of can be written as the ... more

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All the permutations of are nothing but the ... more

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For order , For order , For order , For order... more

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Elements formed by the rotation of square and ... more

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Yes, is a group for composition function ; Apply ... more

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It does not form a group under the stated ... more

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Elements of are formed by bijection or one to one... more

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Apply all four properties to show is the group is... more

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Suppose the corners of pentagon is then elements ... more

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Closure property : Let and be the elements of . ... more

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Suppose , here and are non-zero real numbers ... more

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is non-zero and is real number And hence, is ... more

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then  And hence, satisfies by closure property  ... more

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Multiply both sides by from left And then apply ... more

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For rows, suppose and entries are equal then   ... more

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; Since there can't be a same element in a same ... more

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; Since there can't be a same element in a same ... more

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Since there are three different elements , , in... more

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Let and be an arbitrary elements in which is ... more

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To prove is a group under composition of the ... more

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To prove is an abelian group under composition of... more

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Let .To prove for some positive integer where \... more

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Multiply with from the left of the equality and ... more

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Since, Therefore, Since and Therefore, So it is ... more

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; It can be noted that can be simplified as .... more

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Multiply from the left side of the equality and ... more

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Let which is the same as . Now multiply from the... more

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; Let take a additive group and then take three ... more

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; The group has an element whose order is 2 and... more

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Order is ; Order is ; more

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Check for , ; It's clearly true for . Suppose it ... more

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Hence proved. ; Assuming,  ; more

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True ; Assuming,  ; more

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Hence proved that . ; taking the value of, ; more

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Hence proved ; Since the value of is, ; more

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For the given group G let us suppose is identity ... more

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 eabeeabaabebbea ; Since it is given that G is ... more

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Given : G is a group having elements such that ... more

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Need to prove G is abelian i.e. we will show ... more

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Let us  compute without using any given ... more

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Case1. Let us suppose the group of order  1. Then ... more

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Let us suppose and be elements of G and is ... more

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Let us suppose i.e. where e is identity of G. (... more

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 we will find the unique element x for the given ... more

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Let us consider amd be the order of and . so, ... more

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Let us suppose order of G i.e. for some integer  n... more

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Let us suppose order of and order of is and ... more

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Given that : and Firstly we will show  Let us ... more

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Given Squaring both sidesnow again squaring on ... more

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Using commutative property of multiplication .... more

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Group is abelian. ; Using commutative property of... more

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The set is a group. ; 1. The set is a closed set ... more

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  ; The subgroup has the mapping such as:where ... more

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; The group is of form and hence Hence the ... more

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No elements is possible. ; The group elements of ... more

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; The non-zero group is given by .The operation ... more

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No elements is possible. ; The group elements of ... more

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; The non-zero group is given by .The operation ... more

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The group is generated by set ; The set 11 can be... more

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The sets and generate the additive group of ; In... more

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The additive group is cyclic.  ; The group ... more

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Order of the group is 8. For any two elements ... more

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Since is closed under the group operation so ... more

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According to the theorem 7.11 will be a subgroup ... more

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Let can generate then there will be a for which... more

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Since can be represented as and can be ... more

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Let , since is of finite order and is a element ... more

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Let then will be a subgroup of only if it ... more

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 The element will be in only when there exists ... more

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According to the definition of a cyclic group for ... more

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Let is cyclic group which is generated by the the... more

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This can be proved, if it is proved that for any ... more

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If and are elements in .So, Now, inverse of can... more

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According to inverse element axiom because is a ... more

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Suppose  and hence consists the composition of ... more

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Assume  Hence, is the subgroup of    and then ... more

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is an identity element  hence is proved ; ... more

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with properties  and   consists in the set     ... more

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then  Therefore,  Suppose   consists in     also ... more

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If then All elements of are in group when it is... more

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False all the subgroups of are cyclicthere is no ... more

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According to the theorem when is equal to then... more

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then   And hence product is present in set     ... more

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, then  Then product is present in set     is ... more

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Assume , then Therefore, according to theorem    ... more

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, then  and  product is present in the set    is ... more

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Subgroups of   Subgroups of    ; According to the ... more

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Since every subgroup of a cyclic group is always ... more

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Since the element has order . So the subgroup ... more

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Since is an abelian group, so by commutative ... more

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Proof this statement by contradiction.Suppose is ... more

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Since is an infinite cyclic group, so . To prove... more

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Proof this statement by contradiction.Suppose for... more

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Since is cyclic, so there is some such that . ... more

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To show the given group is cyclic, first apply ... more

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Suppose . Now note that the order of the element ... more

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Let be a non-identity element of .Since is a ... more

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No. ; Suppose is cyclic. Then there is some such... more

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Note that . Now Therefore, the order of all ... more

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The element is the generator of as Therefore, . ... more

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Let . Note that in particular , and that is also ... more

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To show that such that is an isomorphism.By the ... more

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Define a function such that    and,    Clearly, ... more

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To show that defined as is an isomorphism.By the... more

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To show that defined as   is an isomorphism.By ... more

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Since it is , for contains Since distinct ... more

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But alsoDistinct elements have the same labels in... more

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Let are in domain and hence , given function is ... more

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Since it is not given that H is an identity group... more

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For ,  . Since at , given function is not ... more

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Since , it is injectiveSince , given function is ... more

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Since , it is injectiveSince , given function is ... more

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The group contains all the elements which are ... more

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is a homomorphism between , an infinite cyclic ... more

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First, to prove that the function from the ... more

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The following proof is proved by the property of ... more

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and have the same order because is an ... more

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; Let is an arbitrary element in , then there ... more

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Therefore if and only if . ; If then use the ... more

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is homomorphism and , therefore is a subgroup ... more

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; Let be arbitrary elements from , that is and ... more

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; Let . If function determinant is homomorphism ... more

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satisfies the properties of homomorphism, ... more

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. ; Since the set consists of all elements of ... more

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. ; Since the set consists of all elements of ... more

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To show : is a group. For that, show that ... more

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To show : Inn(G) is a subgroup. For that, show ... more

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Suppose be an isomorphism from rational numbers ... more

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and are not isomorphic because is abelian group... more

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  ; In and is not isomorphic for the following ... more

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  ; In and is not isomorphic for the following ... more

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  ; In and is not isomorphic for the following ... more

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  ; In and is not isomorphic for the following ... more

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  ; To solve follow these stepsThe structure of ... more

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  ; To solve follow these stepsThe structure of ... more

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  ; In and is not isomorphic for the following ... more

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  ; In and is not isomorphic for the following ... more

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 First consider a subgroup of that is .The ... more

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If then defined by is bijection of A(G)Then ... more

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If then defined by is bijection of A(G)Then ... more

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To prove that is isomorphic to , first show that ... more

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The group has the elements and group has the ... more

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Find the number of automorphisms from the group ... more

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The number of the automorphisms form to to find ... more

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The number of elements in the automorphism group ... more

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For 3(a) the transposition is . ; For 3(b) the ... more

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This can be expressed as . This is not even. ; ... more

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The set of odd permutation is not a group. The ... more

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There are elements in . The elements and order ... more

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Suppose .. ; If a permutation is even, then the ... more

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Suppose .. ; Order of the element of the group is ... more

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Suppose and be two disjoint cycles in the ... more

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Suppose permutation is written as a product of ... more

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By the theorem, the order of permutation is the ... more

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The set is isomorphic to the group with respect ... more

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The elements in the are of  the form for and .... more

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Let be and and be any in .To prove that  if ... more

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Let and   be the transposition in with .So, and... more

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be the set of all even permutations in .If Then,.... more

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It is given that be a product of disjoint cycles ... more

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Let be a permutation on .To write in disjoint ... more

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To prove every element of is the product of ... more

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To prove every element of is a product of atmost... more

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Let .Then .Let Then there are three cases,Case(1)... more

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Let be the k-cycle.Let .To prove that .Suppose ... more

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Let be the set of all permutations in that fix 1... more

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Define a function where returns the permutation ... more

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As, every element of is even permutation.And, ... more

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Let are transpositions.To prove generates .... more

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Since, .Let and be the disjoint cycle in .To ... more

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Let be an automorphism on defined as for every... more

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Consider as a subgroup of .Define   by.The ... more

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