2. Let G = {a, b, c, d, e} with the operation * defined by the table below. Determine if the system (G, *) is a group. * a b C d a a b C d b b a d C C C d b a d d C a b
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Identify if it is a group
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- Create steps along with justifications to verify that in a group system (G, +) the following property holds:For any two elements from G, ‘a’ and ‘b’, -(a + b) = (-b) + (-a). In other words, the claim is that (-b) + (-a) plays the role of the inverse of a + b. Create steps to show that the element (-b) + (-a) does, in fact, play the role of an inverse to the element a + b, i.e., show that:i. (a + b) + ( (-b) + (-a) ) = e, where e represents the identity in the group; andii. ( (-b) + (-a) ) + (a + b) = e.What is the order of the group ℤ10 x ℤ15 x U(10) /〈(9, 10, 7)〉?Are these sets with the given operation commutative groups? If not, indicate the reason. If several reasons apply, select the first reason that applies. ({0},+) ({-1,1},*) where a*b := a b (ℕ, +) ({1},*) where a*b := a b (ℤ3,*) where a*b := a b
- Show that PSL(2, Z7) = SL(2, Z7)/Z(SL(2, Z7)), which has order 168,is a simple group.If a1, a2, . . . , an belong to a group, what is the inverse of a1a2 . . . an?(a) Suppose that there are 6 people in a room. Show that one can always find a group of 3 people such that either nobody in the group knows anybody in the group or everybody in the group knows everyone in the group.(b) Show that this conclusion does not hold if there are only 5 people in the room.
- Find number of elements of order 2 in A5 (Alternating group)A square has four rotational symmetries (including the identity) and four line symmetries. Make a group table for these eight symmetries. Is this a commutative group?The elements of the multiplicative group G of 33 permutation matrices are given in Exercise 35 of section 3.1. Find the order of each element of the group. (Sec. 3.1,35) A permutation matrix is a matrix that can be obtained from an identity matrix In by interchanging the rows one or more times (that is, by permuting the rows). For n=3 the permutation matrices are I3 and the five matrices. (Sec. 3.3,22c,32c, Sec. 3.4,5, Sec. 4.2,6) P1=[ 100001010 ] P2=[ 010100001 ] P3=[ 010001100 ] P4=[ 001010100 ] P5=[ 001100010 ] Given that G={ I3,P1,P2,P3,P4,P5 } is a group of order 6 with respect to matrix multiplication, write out a multiplication table for G.