OLet a and b be elements of a group G. Prove that G is abelian if and only if (ab)- = a¯\b-!.

Q: Suppose that G is a finite abelian group.  Prove that G has order pn where p is prime, if and only…

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Q: a. Show that (Q\{0}, * ) is an abelian (commutative) group where * is defined as a ·b a * b = .

A: To show this we have to show that this holds closure, associative, identity, inverse and commutative…

Q: Prove or Disprove: If (G, *) be an abelian group, then (G, *) a cyclic group?

A: If the given statement is true then we will  proof the statement otherwise disprove we taking the…

Q: 8. Show that (Z,,x) is a monoid. Is (Z,,x4) an abelian group? Justify your answer

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Q: 9. Prove that a group of order 3 must be cyclic.

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Q: Prove that the fundamental group is abelian if and only if each homomorphism γ∗ as above only…

A: Let us assume that π1(X), the fundamental group is abelian. Let us consider a loop α with…

Q: Show that U5 andZ4 are isomorphic groups?

A: U(5)= {1,2,3,4}, <2> = {2, 22 = 4, 23 = 8, 24 = 1} = U(5) Therefore, U(5) is a cyclic group of…

Q: Prove that a group of order 175 is Abelian.

A: Let G be a group of order 175 We first try to rewrite 175 as prime factorization as follows: 175 =…

Q: 3. Prove that (Z/7Z)* is a cyclic group by finding a generator.

A: Using trial and error method, seek for an element of order 6.

Q: Show that the group of permutations Σ2 is abelian. Then show that Σ3 is not. Writing up the group…

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Q: Show that group U(1) is isomorphic to grop SO(2)

A: The solution is given as follows

Q: Give an example, with justification, of an abelian group of rank 7 and with torsion group being…

A: consider the equation

Q: Prove that a subgroup of a finite abelian group is abelian. Be careful when checking the required…

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Q: Prove that a group of even order must have an element of order 2.

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Q: 7. If x and g are elements of group G, prove that x=g 'xg. Warning: You may not assume that G is…

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Q: The group GLQ,R) abelian group is an

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Q: Prove that a group of order 3 must be cyclic.

A: Given the order of the group is 3, we have to prove this is a cyclic group.

Q: Exercise: Show that (Z5, +5) is an abelian group.

A: The element of Z5={0,1,2,3,4}

Q: Give an example of elements a and b from a group such that a hasfinite order, b has infinite order…

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Q: List the six elements of GL(2, Z2). Show that this group is non-Abelian by finding two elements that…

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Q: Show that ( Z,,+,) is a cyclic group generated by 3.

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Q: Determine all cyclic groups that have exactly two generators.

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Q: find Aut(Z30).  Use the FUndamental Theorem of Abelian Groups to express this group as an external…

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Q: Prove that a group G is abelian if and only if (ab)-1 = a-lb¬ va,bEG

A: We need to prove that a group G is abelian if and only if (ab)-1=a-1b-1 , for all a,b in G. The…

Q: If a, b are elements in an abelian group G, show that (ab)n = anbn  for every positive integer n.

A: It is given that 'a' and 'b' are elements in abelian group G. This means (abn)=(anb).   Let for n=1

Q: (1) Z/12Z

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Q: Prove that a cyclic group with even number of elements contains ex- actly one element of order 2.

A: The solution is given as

Q: Prove that the 2nd smallest non-abelian simple group is of order 168.

A: Introduction- An abelian group, also known as a commutative group, is a group in abstract algebra…

Q: If (G, ) is a group with a = a for all a in G then G is %D abelian

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Q: Prove that a group of even order must have an odd number of elementsof order 2.

A: Given: The statement, "a group of even order must have an odd number of elementsof order 2."

Q: Which abelian somorphic to groups subyraups of Sc. Explin. are

A: Writing a permutation σ∈Sn as a product of n disjoint circles. i.e σ=τ1,τ2,τ3,…τk The order of σ is…

Q: Prove that every subgroup of the quaternion group Qs is normal. Deter mine all the quotient groups.

A: The given group is Q8=±1,±i,±j,±k. There are four non trivial subgroups namely: -1=1,-1.…

Q: Give an example of an infinite non-Abelian group that has exactlysix elements of finite order.

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Q: 2. Prove that a free group of rank > 1 has trivial center.

A: Given:Prove that a free group of rank>1 has trivial center

Q: determine whether the binary operation * defined by a*b=ab gives a group structures on Z. if it is…

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Q: Show that a homomorphism defined on a cyclic group is completelydetermined by its action on a…

A: Consider the x is the generator of cyclic group H for xn∈H, ∅(x)=y As a result, For all members of…

Q: (а) (Q, +) (b) (Zs, ·)

A: (a)(ℚ,+)(b)(ℤ8,.)

Q: Show that there are two Abelian groups of order 108 that haveexactly 13 subgroups of order 3.

A: Aim: There are two Abelian groups of order 108 that have exactly 13 subgroups of order 3.

Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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Q: not

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Q: determine whether the binary  operation * defined by a*b=ab gives group structure of Z. if it is not…

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Q: What is the smallest positive integer n such that there are exactlyfour nonisomorphic Abelian groups…

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Q: ) Prove that Z × Z/((2,2)) is an infinite group but is not an infinite cyclic grou

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Q: Prove that a finite group is the union of proper subgroups if andonly if the group is not cyclic

A: union of proper subgroups  proof: Let G be a finite group. In the first place, we are going the…

Q: 9. Show that the two groups (R',+) and (R'- {0}, -) are not isomorphic. | 10. Prove that all finite…

A: Two groups G and G' are isomorphic i.e., G≃G′, if there exists an isomorphism from G to G'. In…

Q: (a) Give the definition of a gyclic group. (b) Prove that every eyclic group is abelian . (c) Prove…

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Q: Prove that a group G is Abelian if and only if (ab)-1 = a-1b-1 forall a and b in G.

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: Every abelian cy elic O True O False group is

A: to check  whether every abelian group is cyclic or not? proof let a euler group U8=1,3,5,7 let…

Q: The set numbers Q and R under addition is a cyclic group. True or False then why

A: Solution

Q: Use the fundamental theorem of Abelian groups to express Z20 as an external direct product of cyclic…

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