# Simple Group Question

#### conscipost

##### Member
Would it be correct to call mixture of three chemical solutions, namely salt water, salt water with sodium hydroxide, and salt water with HCl, a group? As I understand this, (which is not entirely realistic) mixture of solutions is associative and closed, salt water would be the identity which is yielded from the mixture of the NaOH solution and HCl solution, assuming that all of the NaOH and HCl react to yield NaCl. The only reason I ask, is because mixing a solution with itself yields the same solution, and this is different from other groups like Z3. In the definition of a group I notice that the identity must hold for all elements and since this is not the case and the elements are not simply trivial copies of another, I would say there is only one identity. However, I am still hesitant to accept this example. Thanks for any input.

• Ackbach and Alexmahone

#### Alexmahone

##### Active member
Would it be correct to call mixture of three chemical solutions, namely salt water, salt water with sodium hydroxide, and salt water with HCl, a group? As I understand this, (which is not entirely realistic) mixture of solutions is associative and closed, salt water would be the identity which is yielded from the mixture of the NaOH solution and HCl solution, assuming that all of the NaOH and HCl react to yield NaCl. The only reason I ask, is because mixing a solution with itself yields the same solution, and this is different from other groups like Z3. In the definition of a group I notice that the identity must hold for all elements and since this is not the case and the elements are not simply trivial copies of another, I would say there is only one identity. However, I am still hesitant to accept this example. Thanks for any input.
Basically, this questions asks whether 3 elements: a, b, e with the following properties:

a^2=a
b^2=b
e^2=e
ea=ae=a
eb=be=b
ab=ba=e

form a group.

My answer is no, because it is not isomorphic to $Z_3$.

Simpler reason:
(ab)b=eb=b
a(bb)=ab=e
Associative property is not satisfied.

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• Ackbach and conscipost

#### conscipost

##### Member
Basically, this questions asks whether 3 elements: a, b, e with the following properties:

a^2=a
b^2=b
e^2=e
ea=ae=a
eb=be=b
ab=ba=e

form a group.

My answer is no, because it is not isomorphic to $Z_3$.

Simpler reason:
(ab)b=eb=b
a(bb)=ab=e
Associative property is not satisfied.
That's true. Thanks for pointing that out.
I suppose at the least it is an interesting counter example.

If concentration was considered I can imagine this situation working though.
So, b+b=2b and a+(2b)=b. This would leave the 3 element structure it has now, and I suppose would be isomorphic to (Z,+) where multiples of a are negative integers and multiples of b positive integers.

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