Determining a group, by checking the group axioms

In summary: Since ##1/2-1/(2x)## varies with x, it cannot be an identity.For question 2, I found a G2 identity to be 1/π, and a G3 inverse to be 1/(π²x), (since x does not equal zero). G1 is closed and G4 is associative, so (R*,◦) is a group.In summary, for the given sets with binary operations (R,◦) and (R*,◦), where x◦y=2xy+1 and x◦y=πxy respectively, the first set is not a group as it does not have a single identity element for all elements, while the second
  • #1
umzung
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Homework Statement


  1. For the following sets, with the given binary operation, determine whether or not it forms a group, by checking the group axioms.

Homework Equations


  1. (R,◦), where x◦y=2xy+1
  2. (R*,◦), where x◦y=πxy and R* = R - {0}

The Attempt at a Solution



For question 1, I found a G2 identity to be 1/2 - 1/(2x), meaning x cannot be zero and therefore (R,◦) is therefore not a group. Is this in the right ballpark?

For question 2, I found a G2 identity to be 1/π, and a G3 inverse to be 1/(π²x), (since x does not equal zero). G1 is closed and G4 is associative, so (R*,◦) is a group.[/B]
 
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  • #2
umzung said:
For question 1, I found a G2 identity to be 1/2 - 1/(2x), meaning x cannot be zero and therefore (R,◦) is therefore not a group. Is this in the right ballpark?
It's not so much that x cannot be zero as that the identity varies between elements - i.e. it depends on x. To satisfy the identity axiom (what you are calling G2), the identity must be the same for all elements. That is, it must be a single element e such that, for all ##x##, ##e\circ x=x\circ e = x##.
 

Related to Determining a group, by checking the group axioms

What are the group axioms?

The group axioms are a set of rules that a set and a binary operation must satisfy in order to be considered a group. These axioms include closure, associativity, identity element, inverse element, and commutativity (optional).

Why is it important to check the group axioms?

Checking the group axioms is important because it ensures that the set and binary operation actually form a group. If any of the axioms are not satisfied, then the set and operation do not meet the criteria for a group and cannot be used as such.

Can a set and operation satisfy some, but not all, of the group axioms?

No, in order for a set and operation to be considered a group, they must satisfy all of the group axioms. If even one axiom is not satisfied, then the set and operation cannot be considered a group.

What is the process for determining if a set and operation form a group?

The process involves checking each of the group axioms for the given set and operation. If all of the axioms are satisfied, then the set and operation form a group. If any of the axioms are not satisfied, then the set and operation do not form a group.

Are there any exceptions to the group axioms?

Yes, there are some exceptions to the group axioms. For example, the commutativity axiom is optional, meaning that a group can still be formed even if the operation is not commutative. Additionally, some operations may have unique properties that allow them to satisfy all of the axioms, but are not considered traditional groups.

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