Let H be a subgroup of G. If g e G, show that gHg-1 = {ghg-1|he H} is also a subgroup of G. This subgroup is called a conjugate subgroup of H in G.

Q: Show that if H and K are subgroups of G then so is H ∩ K.

A: Given that H and K are subgroup of group G. We have to show that H∩K is a subgroup of group G.…

Q: Let G be a group and let H and K be subgroups of G so that H is not contained in K and K is not…

A: Given that G be a group and H and K are two subgroup s.t H is not contained in K and  K is not…

Q: Show that if H and K are subgroups of an abelian group G, then {hk|h € H and k e K} is a subgroup of…

A: A set G is called a group if it satisfies four properties  Closure property:  ab∈G  where a,b ∈G…

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: If H is a subgroup of a group G such that (aH)(Hb) for any a, bEG is either a left or a right coset…

A:

Q: Find the right cosets of the subgroup H in G for H = ((1,1)) in Z2 × Z4.

A: Let, the operation is being operated with respect to dot product. Elements of ℤ2=0,1 Elements of…

Q: 10. Let A be a subgroup of G, and let B be a subgroup of H. Show that A×B is a subgroup of G×

A:

Q: 6. If G is a group and H is a subgroup of index 2 in G; then prove that H is a normal subgroup of G:

A: I have proved the definition of normal subgroup

Q: be a group and Ha normal subgroup of G. Show that if x,y EG such that xyEH then yxEH Let G

A: Given: Let G be a group and H a normal subgroup of G.To show that x,y∈G suchthat xy∈H then yx∈H

Q: Let (G, ◊) be a group and x = G. Suppose H is a subgroup of G that contains x. Which of the…

A: Solution of the problem as follows

Q: 32. If H and K are subgroups of G, show that Hn K is a subgroup of G. (Can you see that the same…

A: To show:

Q: Theorem: Let (K,+)is a subgroup of a group (H, ) and (H,) is a subgroup of a group (G,) then (K, )is…

A:

Q: If H is a proper subgroup of L and L is a proper subgroup of G. If |G| = 1250 and |L| = 125, list…

A:

Q: 53. Let H be a subgroup of G and C(H) = {g €G|gh = hg for all he H}. Prove C(H) is a subgroup of G.…

A:

Q: 51. Let N be a normal subgroup of G and let H be a subgroup of G. If N is a subgroup of H, prove…

A: According to our guidelines we can answer only first question and rest can be reposted. Not more…

Q: Let ø be a homomorphism from a group G to a group H. Let K be a subgroup of H. Prove that ø (K) = {…

A:

Q: Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

A: Given that Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

Q: 187. Let G be a finite group and PE Syl,(G). Give an example of a subgroup H of G with HOP not a…

A:

Q: Let G be an abelian group and suppose that H and K are subgroups of G. Show that the following set…

A: According to the given information, Let G be an abelian group and suppose that H and K are subgroups…

Q: The center of G is a commutative subgroup of G.

A: To prove that the center of G is a commutative subgroup of G.

Q: Let G be a group and let H and K be subgroups of G. Prove that the intersection of H and K, H n K =…

A: 1. It is given that H∩K=x∈G|x∈H and x∈K Let  x, y∈H∩K ⇒x,y∈H and x,y∈K⇒xy-1∈H and xy-1∈K⇒xy-1∈H∩K…

Q: . Let H and K be normal subgroups of a group G such that HCK, show that K/H is a normal subgroup of…

A:

Q: f H and K are two subgroups of a group G, then show that for any a, b ∈ G, either Ha ∩ Kb = ∅ or Ha…

A: If H and K are two subgroups of a group G, then show that for any a, b ∈ G,either Ha ∩ Kb = ∅ or Ha…

Q: Let G be a group and let H be a subgroup of G with |G : H| = 2. Prove that H a G, that is, H is a…

A:

Q: . Let H and K be normal subgroups of a group G such nat HCK, show that K/H is a normal subgroup of…

A:

Q: Let M and N be normal subgroups of G. Show that MN is also a normal subgroup of G

A: It is given that M and N are normal subgroups of G. implies that,

Q: Let G be a group and let H and K subgroup of G. Prove that the intersection H and K is a subgroup of…

A:

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 H and K are subgroups of a group G, prove that ANB is a subgroup of G.

A: GIVEN if H and K are the subgroup of a G, prove that  A∩B  is a subgroup of G

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

A:

Q: Let G be a group, H4G, and K < G. Prove that HK is a subgroup of G. Bonus: If in addition K 4G,…

A:

Q: Let H be a subgroup of G and let K=⋂φ∈Aut(G)φ(H). Show that K is characteristic in G

A:

Q: 4. Let H be a subgroup of a group G. Show that exactly one left coset of H is a subgroup.

A:

Q: Let G Są and let K = {1,(1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}. K is a normal subgroup of G. What is…

A:

Q: If H is a subgroup of G, then the index of H in G, written as (G : H), is the number of left (or…

A: Coset of H in G: Let H is a subgroup of the group G Then for any g∈G the set gH=gh : h∈H is called…

Q: Let H and K be subgroups of a group G. Prove that HNK is a subgroup of G.

A:

Q: Let H be a subgroup of G, let a be a fixed element of G, and let K be the set of all ele- ments of…

A:

Q: Let G be a group. Prove that Z(G) is a subgroup of G.

A: The set ZG=x∈G|xg=gx,∀g∈G of all elements that commute with every other element of G is called the…

Q: Let H and K be subgroups of the group G, and let a, b E G. Show that either aH n bK = Ø or else aH N…

A:

Q: Let H and K be normal subgroups of a group G such that HCK, show that K/H is a normal subgroup of…

A:

Q: If H and K are subgroups of a group G then H n K is a subgroup of G.

A: Note:  according to our guidelines we can answer first question and rest can be reposted.   Lemma:…

Q: e subgroups

A: Introduction: A nonempty subset H of a group G is a subgroup of G if and only if H is a group under…

Q: Let H be a subgroup of G. If G has exactly one subgroup of order |H|, then show that for all g e G,…

A:

Q: Let H be a subgroup of G and let g be an element of G. the conjugate subgroup gHg^-1 is defined to…

A: We have to prove that gHg^-1 is a subgroup of G

Q: Let G be a group and let a E Ga G with a = 8. the order of a² is not equal to the order of the…

A: The given statement is

Q: Prove that ifH and K are subgroups of a group G with operation *, Question 8. then HNK is a subgroup…

A:

Q: Prove that if H is normal in G and ø is onto, then ø[H] is normal in G'.

A:

Q: Let H be a subgroup of G such that x^2 ∈ H for all x ∈ G, then show that H is a normal subgroup of…

A: H = {x² : x ∈ G} And, H &lt; G

Q: Let H be a subgroup of G, define C(H) the centralizer of H.

A: