Let H be the set of elements (ª of GL(2, R) such that ad– bc=1. Show that H is a subgroup of GL(2, R). H is called the special linear group of degree 2 over R and is denoted by SL(2, R).

Q: Let A and B be rings. Define addition and multiplication on the Cartesian product A x B by…

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Q: 1) Prove that G = {a + bv2 \a,b € Q and a, b # 0} is a subgroup of (R – (0},.)

A: Subgroup

Q: Let S and T be two involutive transformations of the plane. (a) Prove that S T is involutive if and…

A: (a) If  S and T are involutions, then  I = (ST)2 = ST.ST if and only if  S2T ST2 = SIT = ST That is…

Q: describe all the elements in the cycle subgroup GL(2,R) generated by the given 2 x 2 matrix [0 -1]…

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Q: Find all distinct subgroups of the quaternion group Qs, where Q8 = {+1,±i,±j, £k} Deduce that all…

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Q: 5.3 Let H be the set of elements () of GL(2, R) such that ad-bc-1. Show that H is a subgroup of…

A: be the set of elements  of  such that  that is the We have to show that  is a subgroup of

Q: Let G be a group. a) Define the center Z(G) b) Show that Z(G) is a subgroup of G c) Show that Z(G)…

A: WE HAVE GROUP G  THEN WE HAVE TO DEFINE CENTRE OF G AND SHOW THAT IT IS SUBGROUP AND NORMAL SUBGROUP…

Q: 1. Let 0(V) be the set of all orthogonal transformations on V. Prov O) is a group with respect to…

A: Let O(V) be the set of all orthogonal transformations on V. The determinant of an orthogonal matrix…

Q: Show that the determinant gives a group homomorphism det: GL(n,R)->R*, where R*=R-{0} under…

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Q: Q4: Let A and B be subgroups of a group to that A1 = 4 and 1B1-5- Show that AnB = {e}

A: Let A and B be the two subgroups of Group GSuch  thati.e., the Order of subgroup A is 4 and B is 5…

Q: 3. Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the…

A: Result: Let G, * be a group and H is a subset. The subset H is said to be subgroup, if for every a,…