. Suppose that G and H are groups, and that φ : G → H is a homomorphism. Prove that G/Ker(φ)-φ(G). (The notation įs shorthand for "is isomorphic to.) Definition: Suppose G is a group, and H a normal subgroup of G. The group consisting of the set G/H with operation defined by (aH (bH) (ab) is called the quotient group of G by H. (Sometime the ter factor group" is used in place of "quotient group".) Definitions: . Suppose that (Ga) and (Ho) are groups. We say that a function φ : G → H is a homomorphism if for all a, be G, ф(a b)-ф(a)od(b). Definition: We say that groups are isomorphic if there exists an isomorphism between them. If G and H are isomorphic groups, we write in symbols GH exists an isomorpiism

Q: 5. E Prove that G is an abelian group if and only if the map given by f:G G, f(g) = g² is a…

A: The solution is given as

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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…

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A:

Q: 4. 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: Here we use the properties of groups

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A:

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: 15. Let (G, *) be a group, and let H₁, H₂,..., Hk be normal subgroups of G such that H₁ H₂0... Hk =…

A:

Q: 3. Let G be a group, H a subgroup of G, and g and element of G. (a) Prove that {ghg-1 | h e H} is…

A: Let G be a group, H is a subgroup of G, and g be an element of G

Q: Let G be a group and H, K are subgroups of G with HK=KH. Prove that HK is a subgroup of G.

A: Given that, G be a group and H, K are sub groups of G with HK=KH. Let x∈HK. Then x=hk for some…

Q: Suppose that o: G→G is a group homomorphism. Show that () p(e) = ¢(e') (ii) For every gE G, ($(g))-1…

A:

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

A: We have given a map ,  ϕ : A → B , where A = ℝ , +   ,  B = ℝ* , · such that  , ϕx = 2x We know that…

Q: 2. Let G be a group. Prove or disprove that z= { XE G: xg = gx for all ge G} is a Subgroup of G.

A:

Q: Let f:G-G be a group homomorphism then H = {a € G:f(a) = a} is subgroup O True False

A:

Q: Let H and K be subgroups of a finite group G with H C KC G. Prove that |G:HI |G:K| |K:H].

A:

Q: Let G be a group and g E G. Show that Ø:Z→G by Ø(n) = g" is a homomorphism and isomorphism.

A:

Q: Suppose that p:G→G'is a group homomorphism. Show that () p(e) = ¢(e') (1) For every gEG, (ø(g))-l =…

A:

Q: let G be a group and H a subgroup of G. prove that for any element gEG holds that gH=H if and only…

A: We can solve the given question as follows:

Q: Question 5. Suppose G is an Abelain group, |G| = n < ∞, and |g| # 2 for all g E G. Prove that the…

A:

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

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A:

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

A:

Q: Let G be a group of odd order. Show that for all a E G there exists b E G such that a = b?.

A: Consider the given information, Let G be a group of odd order then, |G|=2k+1 where k belongs to…

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…

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A:

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

A:

Q: Let a e G. Prove that $(a") = ¢(a)" for all n e Z.

A:

Q: Question 5. Let G be a group and H a subgroup of G. For any element g E G, define gH Abelian and |g|…

A: We will prove that K is subgroup of G by using the result. A non empty subset K of group G is…

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 G be a finite group. Let x E G and let G be the conjugacy class of r. Prove that |#C| < |[G,…

A: Given G be a finite group. Let x∈G. Define: xG=gxg-1: g∈G Let G:G=m. Define a map π:G→GG:G defined…

Q: If G is a finite group with |Gl<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A: Given: The group G is a finite group with | G | &lt; 120 and G has subgroups of orders 10, 15, and…

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 G be a finite group. Let E G and let xG be the conjugacy class of x. Prove that x| < |[G, G]],…

A:

Q: Let(G,*) and (H,#) be a groups if f: G H and g: H G are homomorphism such that gof = IG.f og = IH…

A:

Q: Let G be a group, and let xeG. How are o(x) and o(x) related? Prove your assertion

A: According to the given conditions:

Q: 2. Let G be a group. Prove or disprove that Z= {x E G: xg= gx for all g€ G} Isa Subgroup of G.

A: To show Z is a subgroup of G, we need to show that (a) Z is non empty (b) For every a , b∈Z , we…

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: Let (G, 0) be a group and x € G. Suppose H is a subgroup of G that contains x. Which of the…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let G be an Abelian group and H 5 {x ∊ G | |x| is 1 or even}. Givean example to show that H need not…

A:

Q: If G is a finite group with IG|<160 and G has subgroups of orders 10, 16 and 20 then the order of G…

A: Given :- order of a group G... |G| &lt; 160 Also order of subgroups of group G are 10, 16 , 20. We…

Q: dicfin et Prove that a group G has exactly 3 6. - subgroups iff G is a ylic grop ef ender på pis…

A:

Q: Let G be a group with identity element e, and let H and K be subgroups of G. Assume that (i) H and K…

A:

Q: 2. Let o : G → G' be a homomorphism of groups, and let H be a subgroup of G. (a) Let a E G. Prove…

A: Given   ϕ:G -&gt; G' is a homomorphism. H is a subgroup of G. The solution to the subparts is given…

Q: Let F and H be subgroups of group G and let FCH. Prove (G : F) = (G : H)(H : F).

A: It is given that F and H are subgroups of G and F⊂H.

Q: Suppose that 0: G G is a group homomorphism. Show that 0 $(e) = ¢(e') (1) For every gEG,…

A:

Q: 1. (103) Let G be a group with identity element e, and let H and K be subgroups of G. Assume that…

A:

Q: For the homomorphism : D4 → ({1, −1}, •) defined by Þ(a) = { 1 -1 if a is a rotation if a is a…

A:

Q: |Let G and H be groups, and let 0 : G > H be a homomorphism. The set {x E G| 0(x) |e} is called the…

A: From the given information:Let G and H be the groups

Q: Let H and K be subgroups of a group G and assume |G : H| < +co. Show that |K Kn H G H\

A: Let G be a group and let H and k be two subgroup of G.Assume (G: H) is finite.

Q: 2. Let H and K be subgroups of the group G. (a) For x, y E G, define x ~ y if x = hyk for some h e H…

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.