Prove that every group has a unique identity element
Answers
Answered by
2
Prove that identity element is unique
The usual definition of a two-sided identity is that a e = e a = a for each a ∈ G – MPW May 13 '16 at 5:38.
One clear problem is that you have defined a unique inverse prior to defining a unique identity. – CogitoErgoCogitoSum Sep 12 '17 at 22:23.
Hope you will make my answer brainliest please please please.
The usual definition of a two-sided identity is that a e = e a = a for each a ∈ G – MPW May 13 '16 at 5:38.
One clear problem is that you have defined a unique inverse prior to defining a unique identity. – CogitoErgoCogitoSum Sep 12 '17 at 22:23.
Hope you will make my answer brainliest please please please.
Answered by
1
The definition of a group does not require that a−1⋅a=e for every identity element e. It only requires that there be at least one identity for which that is the case.
It says:
There is an identity element e for which e⋅a=a⋅e=a.
Every element a has an inverse a−1 for which a−1⋅a=e.
This does not rule out the possibility that there is another identity element e2, which has the identity property e2⋅a=a⋅e2=a, but for which a−1⋅a≠e2.
We could have a−1⋅a=e≠e2 and the axioms would still be satisfied.
It says:
There is an identity element e for which e⋅a=a⋅e=a.
Every element a has an inverse a−1 for which a−1⋅a=e.
This does not rule out the possibility that there is another identity element e2, which has the identity property e2⋅a=a⋅e2=a, but for which a−1⋅a≠e2.
We could have a−1⋅a=e≠e2 and the axioms would still be satisfied.
Similar questions