Let G = {a + b/2|a, b € Z}. Show that G is a group under ordinary addition.

Q: let G be an abelian group. And let H = {r :z€ G) show that H < G? %3D

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Q: If g and h are elements from a group, prove that ΦgΦh = Φgh

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Q: Let G be a and group Z(G)=< g€G•xg=gx, VX€G} %3D be the center of G. Show that G is commutative if…

A: Solve the following

Q: Prove that the group G = [a, b

A: Given, the group G=a, b with the defining set of relations…

Q: Let G be a group and let a, b E G. (a) Prove that o(ab) = o(ba). (Note that we are not assuming that…

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Q: group ⟨a,b,c|b^4 = a,c^(−1) = b⟩ is abelian.

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Q: Prove that E(n) = {(A, ¤) : A e O(n) and E R"} is a group. %3D

A: Consider the given: E(n)={(A,x)} where A∈O(n)and x∈ℝn

Q: Let G be a group. Show that for all a, b E G, (ab)2 = a2b2 G is abelian

A: To prove that the group G is commutative (abelian) under the given conditions

Q: Is the set Z a group under the operation a * b = a + b – 1? Justify your answer.

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Q: For any group elements a and x, prove that |xax-1| = |a|.

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Q: Show that if H is any group and h is an element of H, with h" = 1, then there is a unique…

A: Given that H is a group and h ∈H  Now,we define a mapping f:Z→H  such that   f(n) = hn for n∈Z For…

Q: Let G be a group and let g, h ∈ G. Show that | gh | = | hg |. Remember that | a | denotes the order…

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Q: Show that each of the following is not a group. 1. * defined on Z by a*b = |a+b|

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Q: Let G be a group, and let a E G.  Prove that C(a) = C(a-1).

A: Given: Let G be a group and let a∈G. then we will prove C(a)=C(a-1) If C(a) be the centralizer of a…

Q: Let G be a group and a E G. Define C(a) = {x € G|ax = xa, for all a E G}. Prove that C(a) < G.

A: A nonempty subset H of a group G is said to be a subgroup of G, if it satisfies the following…

Q: Let G be a group and H a normal subgroup of G. Show that if x,y EG Such that xyEH then 'yx€H-

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Q: Prove that if a is the only element of order 2 in a group, then a lies inthe center of the group.

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Q: 2. Let G be a group. Show that Z(G) = NEG CG(x).

A: Let G be a group. We know Z(G) denotes the center of the group G, CG(x) denotes the centralizer of x…

Q: If (G, ) is a group with a = a for all a in G then G is %D abelian

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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…

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Q: Let G = {a +b 2 ∈ ℝ │ a, b ∈ ℚ }. Prove that the nonzero elements of G form a group under…

A: Note : In the question, it has to be a + b2 ∈ ℝ instead of a + b 2 ∈ ℝ. So, we are solving the…

Q: Consider the group G (x E R]x 1} under the binary operation : ** y = xy-x-y +2 If x E G, then x =…

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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: Consider the group G = {x E R such that x # 0} under the binary operation ху X * y = 2 The identity…

A: First option is correct.

Q: Assume (X,o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) | æ € X, y E Y} and define the…

A: The given question is related to group theory. Given: X , ∘ and Y , ∙ are groups.  Let X × Y = x,y |…

Q: Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = -S, \S| = 3 and (S) =…

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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: Let H and K be subgroups of a group G. If |H| = 63 and |K| = 45,prove that H ⋂ K is Abelian.

A: Given: The H and K are subgroups of a group G. If |H| = 63 and |K| = 45 To prove that H ⋂ K is…

Q: Q:: (A) Prove that 1. There is no simple group of order 200.

A: A simple group is a nontrivial group whose only normal subgroups are the trivial group and the group…

Q: Let G be the set (te, ±a, ±b, ±c) where [ 1 10 Show that G form a Group under multiplication.

A: Group: If a set (finite or infinite) together with a binary operation satisfies the four basic…

Q: If G is a group with identity e and a2 = e for all a ∈ G, then prove that G is abelian.

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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: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: Define * on Q by a +b= qb Is Q a group under *? 210

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Q: Let H be a subgroup of a group G with a, b ϵ G. Prove that aH= bH if and only if a ϵ bH.

A: For the converse, assume a-1b∈H, we want to show aH=bH Let a-1b=h for h∈H. Suppose x∈aH. Let x=ah1…

Q: If G is a non-abelian group of order 8 with Z(G) {e}, prove that |Z(G)| = 2.

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Q: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

A: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

Q: Let G be a group and let x EG of order 23. Prove that there exist an element z EG such that z6 – x.

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Q: 25. Prove that R* x R is a group under the operation defined by (a, b) * (c, d) = (ac, be + d).

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Q: Let G be an abelian group, then (acba)(abc)¯1 is

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Q: let G be a group, a,b E G  such that bab^-1 =a^r , for some  r E N, where N are the natural ones,…

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Q: Let G be a group. V a,b,c d and x in G, if axb=cxd then ab=cd then G is necessarily: O Abelian O Of…

A: Solution:Given    G be a group∀a,b,c,d and x in G

Q: Let G be a group, a E G. Prove that a=a + a < 2

A: Concept:

Q: F. Let a e G where G is a group. What shall you show to prove that a= q?

A: Solution: Given G is a group and a∈G is an element. Here a-1=q

Q: Let G be a group and D = {(x, x) | x E G}. Prove D is a subgroup of G.

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Q: . Prove that the group Zm × Zn is cyclic and isomorphic to Zmn if and only if (m, n) = 1.

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Q: / Let x be anan- empty set, show that whe ther (P(X), n) is a a group of not

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Q: Let G be a group and suppose that (ab)2 = a²b² for all a and b in G. Prove that G is an abelian…

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Q: Show if all primitive transformations of the nonzero form x '= x ,y' = cx + dy d are a group.

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.