(iv) Does there exist a group G such that [G, G] is non-abelian? Give an example, or prove that such a group does not exist.

Q: Prove that a group of order n greater than 2 cannot have a subgroupof order n – 1.

A: Given: To Prove: G cannot have a subgroup of order n-1.

Q: A group G is cyclic if and only if there exists at G such that G={a^|n  ∈ Z}. True or False then why

A:

Q: Prove that a simple group of order 60 has a subgroup of order 6 anda subgroup of order 10.

A: If G is the simple group of order 60 That is | G | =60. |G| = 22 (3)(5). By using theorem, For every…

Q: ii) Does there exist a group G such that G/[G,G] is non-abelian? Give an example, or prove

A:

Q: Let G be a cyclic group with more than two elements: 1. Prove that G has at least two different…

A: Given: Let G be a cyclic group.

Q: Let G be a group. Using only the definition of a group, prove that for each a E G, its inverse is…

A:

Q: 1. Let G be an abelian group with the identity element e. If H = {x²|x € G} and K = {x € G|x² = e},…

A:

Q: Q3\ Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

A: (G, *) be a finite group of prime order   To prove (G, *) is an abelian group

Q: If G is a finite group and some element of G has order equal to the size of G, we can say that G is:…

A: We know that a finite group G is said to be cyclic if and only if  there exist an element in G such…

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

A: consider the equation

Q: give an example of a finite, non-cyclic abelian group containing a container of order 5

A: Take the abelian group G=Z5×Z5 of order 25 whose every element (except identity) is of order 5 and…

Q: Every quotient group of a non-abelian group is non-abelian.

A: (e) False (f) True (g) True Hello. Since your question has multiple sub-parts, we will solve first…

Q: Prove or Disprove that the Klein 4-group Va is isomorphic to Z4.

A: The statement is wrong.

Q: True or False with proof "Any free abelian group is a free group."

A: Given statement is "Any free abelian group is a free group."

Q: Prove that any two groups of order 3 are isomorphic.

A:

Q: Show that if H is any group and h is an element of H, with h" = 1, then there is a unique…

A: Given that H is a group and h ∈H  Now,we define a mapping f:Z→H  such that   f(n) = hn for n∈Z For…

Q: et G be a group with order n, with n > 2. Prove that G has an element of prime order.

A:

Q: Every cyclic group is a non-abelian group.

A: False, all cyclic groups are abelian group. This is true

Q: (a) What does it mean for two groups to be isomorphic?

A: see my solution below

Q: let G be a group of order p^2 where p is prime. Show that every subgroup of G is either cyclic or…

A: Given that G is a group of order p2, where p is prime.To prove that every subgroup of G is either…

Q: QUESTION 6 Prove that a cyclic group G= is Abelian.

A: Given cyclic group G = <a>

Q: Q3\Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

A:

Q: (1) Z/12Z

A:

Q: Prove that if a is the only element of order 2 in a group, then a lies inthe center of the group.

A:

Q: Let G be an abelian group of order 2n, where n is odd. Use Lagrange's Theorem to prove that G…

A: Given :               The group G, which is an abelian group of order 2n, where n is odd…

Q: If G is a cyclic group of order n, then G is isomorphic to Zn. true or false?

A:

Q: Let H be the set of all elements of the abelian group G that have finite order. Prove that H is a…

A: Let H be the set of all elements of the abelian group G that have finite order.  Prove that H is a…

Q: Let G be a group and let Z(G) be the center of G. Then the factor group G/Z(G) is isomorphic to the…

A: We have to check G/Z(G) is isomorphic to group of all inner automorphism of G or not. Where, Z(G) is…

Q: Prove that a group G is abelian if and only if (ab)-1 = a-lb-l Va,bEG Attach Filo

A:

Q: Let G be an Abelian group and H 5 {x ∊ G | |x| is 1 or even}. Givean example to show that H need not…

A:

Q: (i). There is a simple group of order 2021.

A:

Q: Suppose H, and H2 subgroups of the group G. Prove hat H1 N Hzis a sub-group of G. are

A:

Q: Show that the quotient group Q/Z is isomorphic to the direct sum of prufer group

A:

Q: Show that the multiplicative group Z is isomorphic to the group Z2 X Z2 8,

A: We know that if two groups are isomorphic than they have same number of elements i.e. their…

Q: Prove that the alternating group is a group with respect to the composition of functions?

A: Sn is the set of all permutations of elements from 1,2,.....,n which is known as the symmetric group…

Q: Prove that if every non-identity element of a group G is of order 2,then G is abelion

A:

Q: prove that a group G of order p^2, where p is a prime, is abelian.

A: Suppose, G is a group of order p2 where p is prime.

Q: Let G be a finite group of order 4 containing no element of order 4. Explain why every nonidentity…

A:

Q: '. Assume that G is a group such that for all x E G, x * x = e. Prove that G is an abelian group.

A: Consider any two elements a and b in G. So, a,b,ab,ba∈G. Note that I am directly writing the…

Q: (4) subgroup of order p and only one subgroup of order q, prove that G is cyclic. Suppose G is a…

A: Given that G is a group of order pq, where p and q are distinct prime numbers and G has only one…

Q: (a) Let G be a non-cyclic group of order 121. How many subgroups does G have? Why? (b) Can you…

A:

Q: a) Is there any relation between the automorphism of the group and group of permutations? If exists,…

A: An automorphism of a group is the permutation of the group which preserves the property  ϕgh=ϕgϕh…

Q: Prove that if G is an abelian group of order n and s is an integer that divides n, then G has a…

A: G is an abelian group of order n ; And, s is an integer that divides n;

Q: suppose H is cyclic group. The order of H is prime. Prove that the group of automorphism of H is…

A:

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 element of a cyclic group generates the group. True or False then why

A: False Every element of cyclic group do not generate the group.

Q: Let G be a group with order n, with n> 2. Prove that G has an element of prime order.

A:

Q: Prove that: Theorem 3: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if…

A:

Q: QUESTION 14 Find up to isomorphism all Abelian groups of order 18.

A: To find- Find up to isomorphism all Abelian groups of order 18.

Q: Let G be a finite non-abelian simple group and let q be prime, then [G] is

A: It is given that G be any finite non Abelian simple group and q be any prime. We have to determine…