Let (G1, •) and (G2 , *) be two groups and p : G1 – G2 be an isomorphism. Then * O G2 is finite if G1 is finite . G2 might not be abelian even if G1 is abelian. G2 might be abelian even if G1 is abelian G2 is abelian if and only if G1 is cyclic.

Q: Let (G1, ) and (G2, *) be two groups and p: G1- G2 be an isomorphism. Then * G2 is abelian if G1 is…

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Q: 3. Prove that a subset H of a finite group G is a subgroup of G if and only if a. His nonempty, and…

A: We have to prove given property:

Q: A group G is cyclic if and only if there exists at G such that G={a^|n  ∈ Z}. True or False then why

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Q: Let m, n ɛ Z+ such that gcd(m, n) = 1. Suppose there are M and N abelian groups, up to isomorphism,…

A: Given that m,n∈ℤ+ such that gcdm,n=1. Suppose that there are M and N abelian groups, up to…

Q: Let m, n e Zt such that gcd(m, n) = 1. Suppose there are M and N abelian groups, up to isomorphism,…

A: We have M and N are abelian groups of order m and n. so, we know that MN is abelian group.

Q: If G is a finite group and some element of G has order equal to the size of G, we can say that G is:…

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

Q: (c) Prove that if G is a (not necessarily abelian) group, a, b e G, and a² = b² = (ab)² = e, then ab…

A: Use property of group and solve it.

Q: Give a careful proof using the definition of isomorphism that if G and G` are both groups with G…

A: Definition :Let, G and G' be any two group . Then a map φ: G →G' is said to be a homomorphism if φ(a…

Q: 6. Let : (G,*) → (G',') be a group isomorphism, and let a € G. Prove that (a). G is abelian if and…

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Q: The group (Z4 ⨁ Z12)/<(2, 2)> is isomorphic to one of Z8, Z4 ⨁ Z2, orZ2 ⨁ Z2 ⨁ Z2. Determine…

A: Consider the group elements, Here the order of K is 6. Consider the order of group, The order of G…

Q: (G,+) is a finite group such that (a+b)^2 = a^2 + b^2 for all a,b E G. show that G is abelian

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Q: Let (G1, ) and (G2, *) be two groups and p: G1→ G2 be an isomorphism. Then O G2 might be abelian…

A: G2 might not be abelian even if G1 is abelian.

Q: Prove that the trivial G is faithful if and only if G = {1G} representation of a nite group

A:

Q: Let (G1, •) and (G2 , *) be two groups and o: G1→ G2 be an isomorphism. Then * O G2 might be abelian…

A:

Q: is] Let G and H be groups, and let T:G→H_be Isomorphism. Show that if G is abelian then H is also…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Suppose that G is a finite Abelian group. Prove that G has order pn,where p is prime, if and only if…

A: We have given G is a finite Abelian group, then we have to prove that G has order pn,where p is…

Q: Let (G1, -) and (G2 , *) be two groups and o : G1→ G2 be an isomorphism. Then * G2 might be abelian…

A: A one-to-one correspondence φ : G1 → G2 between groups G1 and G2 is called a groupisomorphism if…

Q: Let G be any group and p: G -G, (g) =g1 Vg eG. Then: (a) o is a homomorphism (b) o is a homomorphism…

A:

Q: Let (G1, •) and (G2 , *) be two groups and :G1 G2 be an isomorphism. Then * G2 might be abelian even…

A: we know that,  If two groups are isomorphic, then all the property will be similar i.e. one  is…

Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a = e a^4 =…

A:

Q: Let G be a finite group with no element of order 3, and let a E G. (a) Can 3 divide o(a)? Either…

A: Given that G is a finite group such that ∀a∈G, o(a)≠3 (a) We have to determine whether 3 divides…

Q: let G be a group of order p^2 where p is prime. Show that every subgroup of G is either cyclic or…

A: Given that G is a group of order p2, where p is prime.To prove that every subgroup of G is either…

Q: b. Find all abelian groups, up to isomorphism, of order 720.

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Q: There are two group of order 4, namely Z, and Z2 Z2, and only one of them can be isomorphic to G/Z.…

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Q: Let (G1, •) and (G2, *) be two groups and p: G1→ G2 be an isomorphism. Then * G2 might be abelian…

A:

Q: 8. Prove that if G is a group of order 60, then either G has 4 elements of order 5, or G has 24…

A: The Sylow theorems are significant in the categorization of finite simple groups and are a key…

Q: Let G be a group with |G|=187 then every proper subgroup of G is: * Non cyclic None of these Сyclic…

A: Given that G is a group and G=187 i.e., number of elements in G is 187.  First we have to find the…

Q: Let (G1, ) and (G2, *) be two groups and o: G1 G2 be an isomorphism. The O G2 might not be abelian…

A: Two group are isomorphic the the properties of groups are must be same, i.e,  if one group is an…

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: (a). If G is an abelian group, then (b). If G an abelian group, and group. (c). If o(a) = n, then…

A: This is a problem of Group Theory.

Q: If f: G to H is a surjective homomorphism of groups and G is abelian, prove that H is abelian.

A: As we know that a group homomorphism f:G to H is a map from G to H satisfying:

Q: Let G1 and G2 be any two groups and θ : G1 → G2 be a group isomorphism. Let H1 ≤ G1. Prove that H2…

A:

Q: (a) Prove that every element of Q/Z has finite order. (b) Given two groups (G, .) and (H, *).…

A: This question is related to group theory.

Q: Let S = R\{-1} and define a binary operation on S by a * b = a +b+ ab. Prove that S is an abelian…

A: To show that S is an abelian group, we have to prove all these properties 1) S is closed under…

Q: Let G be a group, and a, b € G. Prove that b commutes with a if and only if b- commutes with a.

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Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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Q: Prove that if every non-identity element of a group G is of order 2,then G is abelion

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Q: 8. Prove that if G is a group of order 60, then either G has 4 elements of order 5, or G has 24…

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering second…

Q: Let (G,*) and (H,*) be finite abelian groups. If G x G = H x H then G=H. Show that they are…

A: Given that, G×G=H×H⇒G=H Since G,* and H,* are both finite abelian groups we get,…

Q: Let (G1, •) and (G2 ,*) be two groups and o: G1 G2 be an isomorphism. Then O G2 might be abelian…

A: Since phi : G1 --> G2 is an isomorphism hence G1 and G2 have same properties .

Q: Let G be a group such that a^2 = e for each aEG. Then G is * Non-abelian Cylic Finite Abelian

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Q: Suppose that G and H are isomorphic groups.Prove that G is abelian if and only if H is abelian.

A: Let G and H be isomorphic groups. So there is bijective map φ:G→H satisfying    φ(ab)=φ(a)φ(b)…

Q: Let (G1, ) and (G2, *) be two groups and p: G1 G2 be an isomorphism. The

A:

Q: Prove that a group G is Abelian if and only if (ab)-1 = a-1b-1 forall a and b in G.

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: 9. Show that the two groups (R', +) and (R' – {0}, -) are not isomorphic.

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Q: Show that if H and K are subgroups of an abelian group G then {hk: h element of H and k element of K…

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Q: Prove that a group is abelian if and only if (ab)-l = a-lb-1 va,bEG Attach File Browse Local Files

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Q: Let (G1, •) and (G2 , *) be two groups and p : G1→ G2 be an isomorphism. Then * G2 might not be…

A: G2 is abelian if G1 is abelian.

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.