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Oct
26
comment Why should we care about groups at all?
@Zaz No worries. While I was cycling home, I thought of an example to illustrate this abuse of notation: strictly speaking, it is wrong to speak of "the ring of integers", since the integers are merely a set. But every mathematician would prefer to take the minute risk of ambiguity to saying "the ring whose underlying set is the set of integers, and where the structure operations are addition with 0 as the neutral element, and multiplication with 1 as the neutral element". Life is just too short.
Oct
26
comment Why should we care about groups at all?
@Zaz Of course a set is not equal to a pair consisting of that set and a binary operation on the set. In algebra, this is a common abuse of notation to refer to a set as "a group" or "a ring" or "a field" or "a vector space", etc. when the underlying operation(s) is/are clear.
Oct
26
comment Why should we care about groups at all?
@Zaz I am afraid I have no intention of bumping this post with a trivial edit that would make the post less readable. My list of examples was intended as purely illustrative, it is not a formal introduction to group theory, and there is no need to introduce heavy notation. In the over 4.5 years that this post has been up for, nobody appears to have had any problems to figure out what the group operation is in each of the examples. I can live with your downvote, I was merely satisfying your request for feedback.
Oct
26
comment Why should we care about groups at all?
@Zaz Those are rings, and in particular they are groups under addition. Since your profile says that any feedback is welcome, here is some feedback: it's usually safer to ask questions when you don't understand something, than throwing confident statements out there and distributing downvotes. You might also like to check peoples' profiles before correcting them on what is or isn't a group.
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Mar
5
revised Maximal Unramified Extension of $\mathbb{F}_p((t))$
deleted 157 characters in body
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Jan
27
comment Abelian torsion group of rational points of an elliptic curve
The standard way of doing this is to compute $E(\mathbb{F}_p)$ for a few primes $p$ of good reduction, and to use the fact that the coprime-to-$p$ torsion of $E(\mathbb{Q})$ injects into $E(\mathbb{F}_p)$
Jan
26
comment Classifying the irreducible representations of $\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}$
You need to do what I labelled as Exercise 2.
Jan
25
comment Classifying the irreducible representations of $\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}$
@SvenWirsing: In the specific situation of the OP, they are 1- and $n$-dimensional. In the general situation of my Edit, they are $($dim $\rho\cdot \#$orbit$_H(\chi))$-dimensional. That just follows from the fact that under induction, the dimension of a representation grows by the index of the subgroup that you are inducing from.
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Nov
28
comment Primes of the form $p=a^2-2b^2$.
@KCd: thanks Keith, fixed it.