QI// Let (H, *) be a normal subgroup of the group (G, *) and we define: G/H={a*H: a € G} and we define on G/H by: (a*H) ® (b*H)=a*b)*H. Prove that (G/H, ® ) is a group.

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

A:

Q: O In the group G=Z%6 , let H (9) and K = (18) %3D List the elements inG / H , G/K and in H/K ,…

A: For G/H : H=0,9,18,271+H ={1,10,19,28}..........8+H={8,17,26,35} For G/K:…

Q: Give an example of a group G and elements x, y E G with o(xy) o(x)o(y).

A: (a) Give an example of a group G and elements x, y∈G with oxy<minox, oy. Let us take the group…

Q: Let G a finite group, g E G such

A:

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: Let G be a group with identity element e. Prove that for any element a in G, (a^2)^-1 = (a^-1)^2.…

A: We are given that G is a group with an identity element e. In group theory, a branch of mathematics,…

Q: (H,*) is called a of (G,*) if (H,*) is a group.

A:

Q: Let G be a group and a ∈ G. Suppose that |a| = 18, find |a^21|. Suppose that |a| = 32, find a…

A: We are given that G is a group, and a ∈ G.A) The order of element a is given as 18. a=18 ⇒a18 = e,…

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

A:

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

A:

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

A:

Q: Let X be a group and we then let x and y be an element of X.  Prove that (x*y)^ -1 = a^-1 * b^-1 iff…

A: Since there are some mistakes in given typed question.question may like "Let X be a group and let…

Q: for the cyclic group of order 16 generated by a , show that a^4 generates an invariant subgroup

A:

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

A:

Q: Let G be a group and H a subgroup of G. H is a normal (or distinguished) subgroup of G if and only…

A: The Center of the group is defined as, C=x∈G | ∀y∈G yx=xy. The given definition 1' is: A subgroup H…

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,…

A:

Q: Consider two groups, G 1 ​ and  G 2 ​ . Group G 1 ​ is defined as {1,a,a^2 ,a^3 } with operation ⋅ ⋅…

A: this basically states the following that any 2 cyclic groups of the same order are isomorphic.

Q: chow that An s a Group with respect, to Cooplesition of functjon.

A:

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

A:

Q: prove: let g be a group, if g is abelian then (ab)^2 = (a^2)(b^2)

A: Given g is an abelian group.

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

A:

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) |…

A:

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

A:

Q: Q/ Let the Cayley group C₁ (S) = C₂ Find the group and the subset S. (3)

A: Introduction: A Cayley graph is a graph that represents a group and its generators. Given a…

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: 10 2) (i) Prove that H = (0, 2, 4, 6, 8) is a subgroup of Z (ii) Prove that H is a normal subgroup…

A: Answer

Q: f:H1 x H2 x ... x Hn ---> H1 + H2 + ... + Hn via h1h2...hn ---> (h1, h2,...,hn

A:

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

A:

Q: Qi/ If (H,*) is collection of subgroups of (G,*) then (U H,*) is subgroup of (G,*) Q2/ In (Z9, +9)…

A: