1. (Q,*) where a*b = `(ab)/(6)` 2. (`R^+` ,*) where a*b = `sqrt(ab)`   Are these sets  group or not? Please show me whether it passed the 4 group axioms. Thanks

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mvcdc | Student, Graduate | (Level 2) Associate Educator

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1. (Q, *), where a*b = ab/6

Let `a, b in Q`. Then `a*b = (ab)/6 in Q` since `1/6 in Q` . [Closed]` `

Let `a, b, c in Q` . Then `a*(b*c) = a*((bc)/6) = (a((bc)/6))/6 = ((abc)/6)/6 = (((ab)c)/6)/6 = ((ab)/6)*c = (a*b)*c` . [Associativity] 

The identity element is 6 - `a*6 = (a6)/6 = a, 6*a = (6a)/6 = a` . The inverse of an element a is `36/a` - `a*(36/a) = (a(36/a))/6 =6` .` `

Hence, (Q, *) where a*b = ab/6 is a group.

 

2. (R+, *), where a*b = sqrt(ab).

Let `a, b, c in R^+` . Then `a*(b*c) = a*sqrt(bc) = sqrt(asqrt(bc))` .On the other hand, `(a*b)*c = sqrt(ab)*c = sqrt(sqrt(ab)c)` .Associativity does not hold, and hence the set is not a group under the defined operation.

 

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