7. A binary operation a*b = (a+b)/3 defined on Real Numbers is_ A. just a groupoid B. a semigroup but not a monoid C. a monoid but not a group D. a group
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- 10. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5: Equivalent Conditions for a Group Let be a nonempty set that is closed under an associative binary operation called multiplication. Then is a group if and only if the equations and have solutions and in for all choices of and in .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.What is the order of the element $(\overline{2}, \overline{9})$ in $Z_{4} \times U_{10}$ is ( $Z_{4}$ is the additive group modulo 4 and $U_{10}$ is Euler group)
- Show that the set {5, 15, 25, 35} is a group under multiplication modulo 40. What is the identity element of this group? Can you see any relationship between this group and U(8)?Show that {1, 2, 3} under multiplication modulo 4 is not a group but that {1, 2, 3, 4} under multiplication modulo 5 is a group.Show that a group G of order par for district primes p and q is solvable .
- (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.5. Verify that (ℤ, ⨀) is an infinite group, where ℤ is the set of integers and the binary operator ⨀ is defined as a⨀ b = a2 + b2determine whether the binary operation * defined by a*b=ab gives a group structures on Z. if it is not group then give the axoim that does not hold