Let G, (Isisn) be n groups and G is the external direct product of these groups. Let e, be the identity of the group G, for each i(1 sis n). Then () For each i, H. = {(e,, e, ... , e X, e;+ 1., e, | x; eG)} is a normal subgroup of G

Q: Let G = Z8 x Z6, and consider the subgroups H = {(0, 0), (4, 0), (0, 3), (4,3)} and K = ((2, 2)).…

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Q: 3 Let N be a normal subgroup of G. Suppose Nn [G,G] = 1. Show that N < Z(G). group. -] Show that…

A: Before we start, let's define the terms:Normal subgroup: A subgroup N of a group G is called a…

Q: Suppose A and B are n×n matrices such that B is invertible and AB is invertible. Prove that A is…

A: Given that the matrices A and B are of order n×n and both A, AB are invertible.

Q: E If (H, *) is a subgroup of the group (G, *). then va e G the pair (a' H *a,*) is a subgroup of (G,…

A: Given below the proof

Q: Qa\ Answer for only one. 1) Let H and K be normal subgroups of a group G, then Hn K be a normal…

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Q: 7. Consider an element z = the order of r is the least common multiple of the orders of its…

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Q: 1. Let G and G' be two groups. Let p: G G' and :G G' be two homomorphisms then prove that H = (x EGI…

A: Using the definition of subgroups, we will first show that H is a subgroup of G. And then we will…

Q: Consider the group G = Z4 X Z9 where the operation is done componentwise, that is (a, b) * (c,d) =…

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Q: Q/ In (Z6 , +6 ) find the cyclic subgroup generated by 1, 2, 5.

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Q: Consider the group G = {x E R such that x 0} under the binary operation x*y=-2xy O x*x*x=4x^3…

A: Multiplication of the elements of the group elements with respect to binary operation

Q: Let us consider that S is subset of group G such that                      S = { x in G ; x^2=e }…

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Q: Let G = (Z;, x,) be a group then the order of the subgroup of G generated by 2 is О а. 6 O b. 3 О с.…

A: We have to find order of subgroup of G generated by 2.

Q: 1) Let G = {e,a,a',a²,a*,a,a°,a'] where a = e and let H = (a*) be a cyclic subgroup %3D %3D %3D of…

A: We have to find quotient group  G/H .

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

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Q: 12. Prove that the intersection of any family of normal subgroups of a group (G, *) is again normal…

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Q: If G is a finite solvable group, show that there exist subgroups of G {e} = H, C H, C H, C · · CH, =…

A: We need to prove that there exist subgroups of G, e=H0⊂H1⊂H2⊂...⊂Hn=G such that Hi+1Hi has prime…

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: Theorem 2. Let G, and G, be groups, then @ Gx G,= G, × G, (6) If H = {(a, e,)| a e G} and H, = {(e,…

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Q: Q3) Prove or disprove 4. Any non-trivial group has at least 2 normal subgroups.

A: Here we have to prove that any non trivial group has atleast 2 normal subgroups.

Q: QI// Let (H, *) be a normal subgroup of the group (G, *) and we define: G/H={a*H: a E G} and we…

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Q: The pressure P and volume V of an expanding gas are related by the formula PV = C, where b and C are…

A: Using the product rule of derivative, it can be written that uv'=uv'+vu'. It is known that…

Q: If N is a normal subgroup of order 2 of a group G then show that N CZ(G).

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Q: If (G,*) is group and each subgroups of (G,*) is normal subgroups then (G,*) is abelian group.

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Q: Q3:(A) Prove that every group of order 15 is decomposable and normal. (B) Show that (H,.) is a…

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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: Define the concept of isomorphism of groups. Is (Z4,+4) (G,.), where G={1,-1.i.-i}? Explain your…

A: Lets solve the question.

Q: 4. a) Let H be the set of elements [a b] of G of GL(2,R) such that ab- bc = 1. Show that H is a…

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

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Q: One of the following is a p-sylow subgroup of the group Z, × Zg H={(0,0),(0,4)} O…

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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: Q1/ If (H,*) is collection of subgroups of (G,*) then (U H,*) is subgroup of (G,*)

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Q: (4) Let G be a group and H ≤ G. The subgroup H is normal in its normalizer NG(H), this imply that…

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Q: Let G be a group and let a be an element of G, with |a| = 30. H = , K = . How many distinct cosets…

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