Let us consider that S is subset of group G such that                      S = { x in G ; x^2=e }  Where e is identity element of G. then prove that S is subgroup.

Q: Let G be an abelian group and let H= {g EG: g^n= e for some nEN}. Prove that H is a subgroup of G.

A:

Q: on G by a * b = ba. Prove that G is isomorphic

A:

Q: Question 7. Prove or disprove: (a) The union of two subgroups of a group (G, *) is a subgroup of G.…

A: 7)  We need to prove or disprove the given statements. a) The union of two subgroups of a group G ,…

Q: Suppose that g and h induce the same inner automorphism of a group G. Prove that h-'g E Z(G).

A: Given: g , h induce the same inner automorphism of a group G. Prove that: h-1g∈ZG.

Q: Exercise 8.3. (a) If H₁ and H₂ are subgroups of groups G₁ and G₂, respectively, prove that H₁ÐH₂ ≤…

A:

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

A:

Q: Let Phi be an isomorphism from a group G onto a group H. Prove that phi (Z(G)) phi Z(H) , (i.e. the…

A: Given that phi is an isomorphism from a group G to a group H.Z(G) denote the center of the group G…

Q: Q2: Let (G,) be a commutative group, and let the set H consist of all elements of G with finite…

A: Given a group G and a set H of G with the given conditions. We need to show that H is a normal…

Q: 1. Show that the symmetry group of F is a subgroup of the isometry group in the Euclidean plane by…

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: ts] Let G be a group, and let H be a subgroup of G. Show that if G is cyclic then H is also cyclic.…

A:

Q: Let ø be a homomorphism from a group G to a group G'. Prove that ker øis a subgroup of G.

A:

Q: Let : G H be a homomorphism. Let (G) = {(g) | ge G}. Show that (G) is a subgroup of H.

A:

Q: A homomorphism f of a group G into a group G' is an ismorphism iff ker f = {e}.

A: We have to proof that a homomorphism f of a group G into a group G' is an isomorphism if and only if…

Q: 10. Let G be abelian and let H be a subgroup of G. Show that G/H is abelian.

A: We have to prove that given theorem.

Q: Show that the center Z(G) is a normal subgroup of the group G. Please explain in details and show…

A:

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: Exercise 8.6. Let G be a group. (a) Prove that G = {e} ≈ G. (b) Prove that G/{e} ≈ G. (c) Prove that…

A: 8.6 Let G be a group   (a) To Prove: G⊕e≅G (b) To Prove: G/e≅G (c) To Prove: G×e≅G

Q: a The group is isomorphic to what familiar group? What if Z is replaced by R?

A:

Q: Let N be a normal subgroup of a finite group G.  Prove that the order of the group element gN in G/N…

A:

Q: (a) If H and K are subgroups of a group G, prove that HK is a subgroup of H and a subgroup of K. b)…

A: a) Let  S be a subset of a group G. Then S is subgroup of G iff i) for a,b∈S ⇒ab∈S ii)for a∈S⇒ a-1∈S

Q: Let G be a group and H, K≤ G normal subgroups of G. Prove HnK≤G.

A: Given that  is a group and  are normal subgroups of .We need to prove that  is a normal subgroup.We…

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

A:

Q: f f is a homomorphism of group G into a  group G' with Kernal K, then K is a normal  subgroup of G.

A: If f is a homomorphism of group G into a group G' with Kernlal K, then we have to prove that K is a…

Q: 2) Let G be a group and H be a subgroup of G then H x = H• y -y-. xcH. true O false

A: (a) Given that G is a group and H is a subgroup of G H.x=H.yH.x.y-1=H.yy-1H.xy-1=Hxy-1∈H Hence,…

Q: Given a group hG, ∗i and a subset H ⊂ G. Prove that H is a subgroup of G if (i) H 6= ∅, and (ii) if…

A:

Q: Prove that every subgroup of the quaternion group Qs is normal. Deter mine all the quotient groups.

A: The given group is Q8=±1,±i,±j,±k. There are four non trivial subgroups namely: -1=1,-1.…

Q: G = (R, +), H = {a+bv2: a,b € Z}

A: Given G = (R, +), H = {a+b√2 : a, b∈Z}. We check whether H is a subgroup of G.

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: 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…

A:

Q: prove that the left and right cosets of the center of a group are equal in order to show that the…

A:

Q: G

A: According to the question, Let's start by defining the dihedral groups D3 and D5,  So, D3 is the…

Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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: Give an example of a group G and subgroup H ≤ G such that ♬ is not normal in G.

A:

Q: 19. Let H and K be subgroups of a group G. Prove that gH ngK is a coset of HOK in G.

A: