Abstract Algebra

J

JaredO

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I was given a question for homework and I cannot figure out where to start.
I know commutativity is involved, however I am not sure how to use the cardinality of the generator to show it. I would appreciate any help!image.jpg
 
Jomo said:
Which elements are in H? Let's start from there.
What are the requirements for H to be a subgp of G?
Click to expand...
A subgroup needs to be closed under the operation (which is the main part confusing me), have the identity of G, and for an a to exist in H, the inverse of a is in H
 
So is the id of G in H? Why, ie prove it.
Suppose x and y are in H. So why is xy in H? Why are you confused about this? You do know that (ab)-1 = b-1a-1??

You also never really answered my 1st question. In your mind, how does x in G get to be in H?????
 
JaredO said:
I was given a question for homework and I cannot figure out where to start.
I know commutativity is involved, however I am not sure how to use the cardinality of the generator to show it. I would appreciate any help!
Click to expand...
To JaredO: you probability don't know this but there is less standard notation in abstract algebra that most other areas of mathematics. Therefore, when posting a question you needs to define terms. What does \(\displaystyle |\left<{\bf{a}}\right>|\) mean?
 
pka said:
To JaredO: you probability don't know this but there is less standard notation in abstract algebra that most other areas of mathematics. Therefore, when posting a question you needs to define terms. What does \(\displaystyle |\left<{\bf{a}}\right>|\) mean?
Click to expand...
I was assuming it is the cardinality of the generator set.
 
JaredO said:
I was assuming it is the cardinality of the generator set.
Click to expand...

I think you mean, the cardinality of the subgroup generated by a. Here, a is the generator of the subgroup, so the "generator set" would be {a}.

See https://en.wikipedia.org/wiki/Generating_set_of_a_group

Now, can you more fully answer Jomo's questions, showing your attempts at proving the various parts of the definition of a subgroup?
 
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