Homework Statement
If \phi : S \to T is a semigroup morphism then show that if a\; \mathcal{R} \;b in S then a\phi \; \mathcal{R} \; b\phi in T.
Homework Equations
Recall that if S is a semigroup then for a\in S aS = \{as : s \in S \}\text{,}\;\;\;aS^1 = aS \cup \{a\}\text{.} The...
Homework Statement
Consider the irreducible representation V in the symmetric group S_5 corresponding to the Young diagram (these are meant to be boxes): [\;\;][\;\;] \\ [\;\;][\;\;] \\ [\;\;]\;\;\;\;
(a) List all standard Young tableaux of the given shape (that is, list all the possible...
Say we have the symmetric group S_5.
The permutations of \{2,5\} are the identity e and the transposition (25).
But what are all the permutations of \{3\}? Is it e and the 1-cycle (3)?
Is it true that if A \equiv B \mod{\varphi(N)} where \varphi (N) is Euler's totient function then a^A \equiv a^B \mod{N}?
I'm not after a proof or anything but I didn't do a number theory course and it seems that this fact is used in many questions I'm currently doing.
So, assuming t\neq 0 (for if t=0 the equation is trivially true), I can conclude that the equation will be true for \alpha =1 for all c\in\mathbb{C}, a=0, b=0 and when \alpha \neq 1 it will only be true for a,b,c=0?
In other words, whatever the value of \alpha, the equation will be true for...
If \alpha =1 then a=0 and b=0
If \alpha \neq 1 then the first equation implies a(1-\alpha) = 0 so a=0 since \alpha \neq 1.
Subbing a=0 into the second equation gives b=\alpha b so b(1-\alpha)=0 so b=0 since \alpha \neq 1.
Subbing a=0, b=0 in the third equation gives c=0
x is a variable and t\in\mathbb{R} and \alpha is a fixed constant. We want the equation to be true for all t.
If you equate coefficients you get:
a=\alpha a
2ta+b = \alpha b
at^2 + bt + c = \alpha c
For what values of a, b and c are these true?
Homework Statement
For which values of a,b,c\in\mathbb{C} is the following equation true? a(x+t)^2 + b(x+t) + c = \alpha(ax^2 + bx + c) where \alpha is some scalar.
The Attempt at a Solution
How do I go about this?
Homework Statement
I think I've done (a) and (b) correctly (please check).
I'm stuck as to how to describe all subspaces of V that are preserved by the operators \varphi(t) and how to prove that \varphi can't be decomposed into a direct sum \varphi = \varphi |_U \oplus \varphi |_{U'}...
If V is a 3-dimensional Lie algebra with basis vectors E,F,G with Lie bracket relations [E,F]=G, [E,G]=0, [F,G]=0 and V' is the Lie algebra consisting of all 3x3 strictly upper triangular matrices with complex entries then would you say the following 2 mappings (isomorphisms) are different? I...
Is the polynomial f(x) = x^7 + (3-i)x^2 + (3+4i)x + (4+2i) \in \mathbb{Z}[i][x] irreducible?
2+i is a Gauassian prime isn't it? And 2+i does not divide 1, 2+i | 3-i , 2+i | 3+4i , 2+i | 4+2i and (2+i)^2 = 3+4i which does not divide 4+2i.
My statement of Eisenstein's criterion is the following:
Let R be an integral domain, P be prime ideal of R and f(x) = a_0 + a_1x + ... + a_n x^n \in R[x].
Suppose
(1) a_0 , a_1 , ... , a_{n-1} \in P
(2) a_0 \in P but a_0 \not\in P^2
(3) a_n \not\in P
Then f has no divisors of...
Homework Statement
Homework Equations
Monotone Convergence Theorem:
http://img696.imageshack.us/img696/5469/mct.png
The Attempt at a Solution
I know this almost follows from the theorem. But I first need to write \displaystyle \int_{I_n} f = \int_S f_n for some f_n in such a...