Chris Leary
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 Apr15 comment Subgroups of $\mathbb{Z}_n$ (integers mod $n$) Following up on what @quid said, $m$ and $n$ will generate the same subgroup of $\mathbb{Z}_{18}$ if and only if $\gcd(m,18)=\gcd(n,18)$, which you may verify by listing all the cyclic subgroups of $\mathbb{Z}_{18}$. You may restrict each of $m$, $n$ to be in $\mathbb{Z}_{18}$ of course. Feb9 comment Non abelian subgroup of a abelian group. You can have abelian subgroups in a non-abelian group, but not non-abelian subgroups in an abelian group. Jan18 comment is complex number under absolute value a group? Is the number 0 allowed? If so, what is its inverse? Dec29 comment Why rationalize the denominator? @Ahaan - +1 for the answer. Nonetheless, a good teacher, at any level, should have no trouble realizing that your three radical expressions represent the same number. However, having said that, and being a US resident, maybe not. Dec29 awarded Nice Answer Dec22 answered Removable discontinuity question Oct29 awarded Yearling Sep27 answered Finding a basis for a subspace in $\;\Bbb R^4\;$ Sep11 comment Cant understand some definitions of abstract algebra, can you help me please? I don't know the reason for the term "over", but it is used to indicate the field from which the scalars for the scalar multiplication are chosen. In a similar vein, if one talks about the finite dimensional space $F^n$, then one considers this space as being "over $F$" in the sense that the $n$-tuples have entries that come from $F$. Aug31 comment A Book for abstract Algebra I'm currently using Pinter's book in my abstract algebra course because it is well-written, inexpensive, and has great exercise sets that break sometimes difficult discussions into manageable pieces. Aug24 comment Does every group have a 'cyclization'? @Alexander - It certainly seems every group has a trivial one, which is why I don't think the trivial case is very interesting. I'm kind of distracted by other obligations at the moment, but it seems that $\mathbb{Z}$ has both itself and the trivial group as cyclizations. Aug24 comment Does every group have a 'cyclization'? It might make a better question to ask what groups have nontrivial cyclizations, since not all have non-trivial ones. Aug24 comment Does every group have a 'cyclization'? @Bryan - I assumed you wanted a non-trivial cyclization. Aug24 answered Does every group have a 'cyclization'? Aug2 comment Convergent or divergent $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n^2+1}}\left(\frac{n}{n+1}\right)^n$? @Nick - One thing to remember with textbook answers is that it is not usually possible, because of space limitiations, to give complete, rigorous answers. So, you get the telegraphic version that says "essentially", i.e., asymptotically, your series is a constant multiple of a divergent series, the harmonic. Jul27 comment If one number is thrice the other and their sum is $16$, find the numbers One number is $x$, the other $3x$. They add to 16, so ... . Jul2 awarded Curious Jun30 comment Is Matrix $A^2$ invertible if $A$ is invertible? Follow Thomas' advice. And what do you know about the inverse of a product of invertible matrices? Jun25 comment Is it true that “there is no such thing as the square root of minus one”? @Andre Nicolas Our mathematical conventions are so economical. Perhaps we subconsciously apply Occam's (or, Ockham's) Razor in making these decisions. Jun25 comment How is $\mathbb{F}_4$ generated? @Omnomnomnom - Didn't mean to repeat you. I must have been composing my comment while you were posting.