Does the identity element in group theory have to be a constant and unique?

Answer 1

Yes, the identity element of a group is unique (this proof is in most algebra books-assume there is another and arrive at a contradiction).

I am not sure what you mean by constant. If you mean that it is unique, then see above. If you mean a number, then you should know that groups can be made arbritrarily and so not necessarily based on the real numbers.

Answer 2

yes. The defintion requires it... otherwise algebra would be imposible in a group with a non consntan identity element. How would you solve x - y = 0 in such group if you don't know that the "1" multiplying x is the same "1" multiplying y.

Answer 3

Yes

Answer 4

Yes and here's why. Suppose two elements g and h were both identities. Then since g is the identity, h=gh. Since h is the identity, gh=g. Put these together and you get h=gh=g, so they are actually the same element.

Answer 5

i imagine I sorta were given what you propose. In mathematics there are 2 kinds of communities: •gadgets •multisets the former's definition obviously states that for it to be such, each and each and every of the elements in it must be distinct or in case you'll unique. The latter's definition states otherwise, because the letter itself isn't something yet an insignificant extension of the former. desire this helped!

Answer 6

yes

Answer 7

yes, it does