Proof: Identity Element of a Group is Unique | Abstract Algebra

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The identity element of a group is unique. We prove this ins today’s video abstract algebra lesson.

This is one of the first results you’re likely to prove in an abstract algebra course. It is simple but fundamental, and lets us rest easy knowing something we’d like to be true is indeed true.

We prove the result directly. Let e and f be identities of the group (G, *). Then e*f = e by definition of identity and e*f = f by definition of identity. Hence, e = f and the identity is unique.

We see that when we give two different names to identities of a group, we are actually naming the same element twice, the unique identity element of the group!

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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The outro music is by a favorite musician of mine named Vallow, who, upon my request, kindly gave me permission to use his music in my outros. I usually put my own music in the outros, but I love Vallow's music, and wanted to share it with those of you watching. Please check out all of his wonderful work.

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You are AMAZING- I am confident that I speak for every undergrad, when I say: Thank goodness for people like you and Thank goodness for Youtube- I have no idea how undergrads of a previous generation coped without this immense help - You are a STAR ✨

jdmc

I really appreciate your work. You explain concepts that are difficult to understand in a very simple and detailed way that make someone appreciate the beauty of mathematics. Kindly consider helping us understand measure theory.

presleykiplagat

Bro, this helped me a lot thank you !

xyz-fqlv

Can I prove it by saying for any arbitrary group element a the following equation holds: a*e1 = a, hence, e1 = a*a^-1; the same goes for e2 = a*a^-1, therefore e1=e2

uzferry

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