Any proper subgroup S of a nilpotent group G is a proper subgroup of its normalizer N_{G}(S) in G.

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

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Q: Explain Special Linear Group is a Normal Subgroup of General Linear Group?

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Q: Let a group G be the internal direct product of two  its subgroups H\index{1} and H\index{2}. Then…

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Q: Find all abelian groups (upto isomorphism) of order 3375.

A: We have to find the all abelian groups (upto isomorphism) of order 3375.

Q: Show that the group of automorphism of a cyclic group of degree four is of order two.

A: Claim :  The group of automorphism of a cyclic group of degree four is of order two.

Q: prove that the group G=[a b] with defining set of relations a^3=e, b^7=e, a^-1ba=b^8 , is a cyclic…

A: We need to prove that , group G = a , b  with defining sets of relations a3 = e   ,  b7 = e  also…

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

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Q: If cyclic group G is generated by an element  a of order n, then a\power{m} is a generator of G iff…

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Q: (a) (Z4, +). (b) (Z5, .). (c) (Z5, +). (d) (Z4, .). (e) (Z6, .).

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Q: Give an example of a non-Abelian group that has exactly threeelementsof finite order.

A: Let G=Z3⊕Z be an additive group where e=0,0. Let x∈G then it can be observed that x=a,b where a∈Z3,…

Q: does the set of polynomials with real coefficients of degree 5 specify a group under the addition of…

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Q: Consider the group G = {x € R such that x* 0} under the binary operation x*y=-

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Q: 2. If H and K are normal subgroups of a group G and H^K = {e}, prove that G is isomorphic to a…

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Q: Which of the following subgroup of S_3 is not normal? Improper subgroup A_3 None of them Trivial…

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Q: Let G be a finite group of order n, positive integer, with identity e. Prove or disprove: For any g…

A: Let G be a finite group of order n, with identity e i.e, order of (G) = |G| = n .   Now let g be any…

Q: Determine the group Z42/ and find a group that it is isomorphic to it. Generalize your observation.

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Q: Let G be a group with order 8. If a is an element of G such that the order of a is 4, find the order…

A: The order of an element a in a group G is the smallest positive integer n such that an=e, where e is…

Q: Evaluate the following integral over the Region R. (Answer accurate to 2 decimal places). [ [. R =…

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Q: *Prove: Every subgroup of a cyclic group is cyclic.

A: Let G be a cyclic group generated by an element a. That is, G = ⟨a⟩. We want to prove that every…

Q: G group is called meta- Abelian if it has an Abelian subgroup N which is normal in G and is Abelian.…

A: Given that We say a group G to be meta- Abelian if it contains an abelian subgroup denoted by N…

Q: Let K be a subgroup of G and let H = {g : gKg^-1} = K. Show that H is a subgroup of G

A: Let K be a subgroup of G and H be defined as H = { g : gKg-1 = K }. Then, we have to show that H is…

Q: Find all abelian groups (upto isomorphism)  of order 67

A: We have to find all abelian groups (upto isomorphism) of order 675.

Q: Let D_8 = . Let S_3 be the symmetric group of degree 3. Let G = D8 x S3. Let H = x and K = x…

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Q: Definition 4.24: Let G be a group, and let Ω be the set of subgroups of G. Let G act on Ω by…

A: We prove our claim by using definition of NG(H).

Q: Let G be a group with the operation * and let H be a subgroup of G. Show that if H and G/H have the…

A: To show that H is a normal subgroup of G, we need to demonstrate that for every element g in G, the…

Q: Prove that if H is a subgroup of a finite group G of index p, where p is the smallest prime that…

A: This question is about group theory.

Q: Let a and b be elements in a group G. Prove that ab^(n)a^(−1) = (aba^(−1))^n for n ∈ Z.

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Q: Define a 'characteristic subgroup' of a group. Also give an example of a subgroup H, of a group G,…

A: A characteristic subgroup is a subgroup which remains fixed by all automorphisms of $G$. We give the…

Q: your a) G = (Z12,O); H = {0,2,4, 6, 8, 10} %3D

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