ab Define * on Q* by a * b =: Show that (Q*,*) is a group. 1.1 1.2 Let a, b be elements of a group G. Assume that a has order 5 and a'b = ba³. Prove that ab = ba. Let G be a group and Z(G) = {a e G: ax = xa for all x E G}. Show that Z(G)is 1.3 a normal subgroup of G. Let G be a group and let p:G → G be the map p(x) = x-1. (a) Prove that p is bijective. (b) Prove that p is an automorphism if G is abelian. Let a, ßE S, (Symmetric group), where a = (1,2)(4,5) and ß = (1,6,5,3,2). Verify that (aß)1 = B-1a-1 . 1.4 1.5

Q: Let p be an odd prime number. Let Dp be the diehedral group of order 2p. Find |Aut (Dp)|.

A: Dp is the Diehedral group of order 2p, where p is an odd prime number.  We have to find AutDp.

Q: et (G, *) be a group. Show that (G, *°P), where x *ºp y = y*x (x, y = G), is a group.

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Q: 9. Describe the group of the polynomial (x* – 1) e Q[x] over Q.

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Q: group , Poq EG, and p"qp=q? + q! pq=p's prove that Let G be a

A: I have done algebraic calculations.

Q: Q1) Consider the group Z10X S5. Let g = (2, (345)) € Z10X S5. Find o(g). T LOV

A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…

Q: prove and that NOXMALIZER STABLIZERS of CENTRALIZERS" are Sub group 67 groupt

A: To prove that normalizer is a subgroup. The normalizer is defined as N(H)=x∈G:x-1Hx=H. Take, x=e,…

Q: 7. Let (G, *) be a group and let a G. Define C(a)= {bEG: ab ba Z(G) = {bEG: ab-ba for all a € G}…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

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: 7. Let G be an infinite cyclic group. Give an example of a subgroup H in G such that H {e} or G. 8.…

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Q: Show that (Q*,-) is not cyclic..

A: We use the basic definitions of cyclic groups to solve this problem. The detailed solution is…

Q: Q1 Let g be an element of a group G such that r = g and r² = e. Then solve for r in terms of g. Q 2…

A:

Q: 16. Let G be a group and let g and h be in G. Show that . - as defined in #15.

A: Given: G is a group, g,h∈G, ϕg:G→G is a function defined as ϕg(x)=gxg-1 To prove: ϕg∘ϕh=ϕgh Let x∈G…

Q: Given that A and B is a group. Find out if : A→B is a homomorphism. If it is a homomorphism, also…

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Q: Q1)) prove or disprove ( 1. Let (G, *) be a group, if x*y = y*x then (x*y)" = x" * y". 2. Each…

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Q: 2. In each case determine whether the two given groups are isomorphic. Justify your answer. a) (2Z,…

A: a) Given that the groups are 2ℤ,+ and 3ℤ,+.The function is given by φ:2ℤ→3ℤ and can be defined as…

Q: 5.3 Let H be the set of elements () of GL(2, R) such that ad-bc-1. Show that H is a subgroup of…

A: be the set of elements  of  such that  that is the We have to show that  is a subgroup of

Q: 10, Let (G, *) be a group and a, b, c E G. (a) Prove if a *b = a * c, then 6 = C (b) Prove if b * a…

A:

Q: = n and o(b) = m Let G be a group and suppose a, b E G satisfy ab = ba. Assume that o(a) : with (n,…

A: Given G is a group, such that a, b∈G satisfy ab=ba. Also, we have oa=n and ob=m with n, m=1.   To…

Q: We have a semigroup (M, *), where M = R- {k} a a b = a + b-kab. Find also that, (M, *) is a group.…

A: Semi group will be a group if there exist an identity element and every element has inverse.

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

A: Here we have to prove that G is an abelian group.

Q: Let (G, *), (G', *' and (G", ") be groups, and let :G G' and : G' → G" be isomorphisms. Prove that t…

A: We will use the definition of isomorphism to prove that the composition of two isomorphisms is again…

Q: 3) Let G and H be two groups with e and e' as identities. Define P = G x H by P = G x H = { (g,h) |…

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Q: is a group with identity (eg, eH).

A:

Q: 6. Let G be GL(2, R), the general linear group of order 2 over R under multiplication. List the…

A:

Q: Given that A and B is a group. Find out if þ: A→B is a homomorphism. If it is a homomorphism, also…

A: Group homomorphism is nothing but a function defined between two groups. The function must be closed…

Q: 2.10 Let G be the set of all 2×2 matrices ( .), where a, b ER and a² + b² +0. Show that G forms a…

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Q: (a) Consider the solid bounded above by the cone z = Vx² + y? on the sides by the cylinder x² + y?…

A: Answer for sub question a: The given region is bounded below by z=0. Further at the top the region…

Q: Let G be a group and let a, be G such that la = n and 6| = m. Suppose (a) n (b) = (ea). Prove that…

A: According to the given information, let G be a group.

Q: Let G be a group with operation * and e be the identity element of G. Show that for any element a in…

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Q: 5. Let G be a group and define z" z * *** *z for n factors c, where z E G and n E Z+. (a) Suppose…

A: Given that G is a group. Also zn is defined as follows. zn=z*z*z*...*z (a) Use mathematical…

Q: 5. Let (G, *) be a cyclic group, namely G=(a). Prove that (a). if G is finite of order n, then G…

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Q: Let G be a group of odd order. a) Prove that if x E G, then (x2) = (x). b)…

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Q: 2. Suppose M and D are isomorphic groups. Prove that Aut(M) is isomorphic to Aut(D).

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Q: 4. Prove that the set H = nEZ is a cyclic subgroup of the group GL(2, R).

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Q: 3. Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the…

A: Result: Let G, * be a group and H is a subset. The subset H is said to be subgroup, if for every a,…

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

A: