Lemma 8.1 Suppose that X = (X,*) is a group and suppose a E X has finite order n > 1. Ifr, s € Zn are such that r < s then a" ‡ as. If m is any integer, then there exists a unique integer r E Zn such that am = ar. 1. 2.

Q: 21. Suppose a is an element of order rs in an abelian group G. Prove that if r and s are relatively…

A: We will prove this by using the concept of cyclic subgroups and the properties of abelian groups.…

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

Q: Exercise 3.5.2 Suppose that G, G' are finite groups and T:G→G' is a group homomor- phism. Show that…

A:

Q: (Z, +) is a group and infinite group

A: Let a binary operation '*' defined on a set G, then it forms a group (G,*) if it holds the following…

Q: Let A = {x E R|–1< x < 1}. Define an operation * on A as follows: a * b = a+b 1+ab where a, b E A.…

A:

Q: Let G be a group of order 27. Let x = G. Prove that x³ € Z(G).

A:

Q: . Let H be a subset of a group G and a E G. Prove that if G is abelian, then aH = Ha for all a E G.…

A: Let H  be a subset of a group G  and We have to prove if G  is abelian group, then.We know that.As G…

Q: föř áll å, 6 in 8. Prove that a group G is Abelian if and only if G" 9. In a group, prove that…

A: We have to prove that G is a group. Using property- square of every element of group is an identity…

Q: 3. Let n eN be given. Is the set U = {A: det A = ±1} C Matnxn(R) a group under matrix multipli- %3D…

A: By using properties of group we solve the question no. 3 as follows :

Q: 3. Let G be a group of order 8 that is not cyclic. Show that at = e for every a e G.

A: Concept:

Q: 3.4. Let G, H be groups, and assume that G H x G. Can you conclude that H is trivial? (Hint: No. Can…

A:

Q: 4.7 Let G= be a cyclic group of order 24. List all the elements in G that are of order 4.

A: Every element of G can be written as xa for some a between 0 and 23 (including both).  Suppose that…

Q: 1. Define x*y over R\ {-1}by x*y = x + y +xy. Prove that this structure forms an abelian group.

A:

Q: (c) If K is a subgroup of G, then p(K) is a subgroup of H. Given: K < G (d) If K' is a subgroup of…

A: I have mentioned the test used to prove a subset to be a subgroup, as a note above. You can skip it…

Q: (5) Let (G, *) be a group with a² = e for every a € G. Show that G, a*b = b* a (that is, show that…

A:

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

A: 2) S=R∖-1 binary operation defined by a*b=a+b+ab

Q: 10.5 Let G be a group of order 8 that is not cyclic. Show that a= e for every a EG.

A:

Q: 27. Prove or disprove that each of the following groups with addition as defined in Exer- cises 52…

A: Let G = Z2xZ4  i.e G = { (0,0),(0,1),(0,2)(,0,3),(1,0),(1,1),(1,2),(1,3)}  Order of G = 8

Q: 6. Show that the special orthogonal group is compact. SO(n) := {A = O(n)| det A = 1}

A: To show that the special orthogonal group SO(n) is compact, we need to demonstrate that it is both…

Q: orthogonal group O(n) = {A E GL(n, R)| AA = A'A

A: