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RSS Abstract Algebra on Math.StachExchange.com

  • killing form is positive definite on cartan subalgebra
    I am studying Theorem 10 on page 147 of Jaboson's Lie algebra book, and at the end of the proof I believe he uses the fact that the restriccion of the Killing form to $H$, the Cartan subalgebra is positive definite, so my question is: If L is a semisimple Lie algebra over an algebraic closed field $F$ of characteristic 0 and $H$ is a Cartan subalgebra, […]
  • How do you prove the logarithm of a product property using mathematical induction?
    I need help proving the logarithm of a product property using mathematical induction. I'm not sure where to go after coming up with the inductive hypothesis logx_1+logx_2+...logx_n = logk.
  • Principal ideal over a ring
    If we take the ring $k[x,y,z]/(xy-z^{2})$ and we consider the ideal $I=(x,z)$, is the image of $I$, in the ring $(k[x,y,z]/(xy-z^{2}))_{(x,y,z)}$ principal? Thank you for your time.
  • Generators for ideals in Z[x]
    Consider the integral domain ${\bf Z}[x]$ the ring of polynomials with integral coefficients. Prove that for every positive integer $n$, ${\bf Z}[x]$ contains an ideal which has precisely $n$ generators and cannot be generated by fewer than $n$ elements. The case $n=2$ follows from considering the ideal $(2,x)$. This question was answered in the proof that $ […]
  • Suppose that H is a subgroup of a group G and |H| =10. If a belongs to G and a^6 belongs to H [on hold]
    Suppose that H is a subgroup of a group G and |H| =10. If a belongs to G and a^6 belongs to H, what are the possibilities for |a|?
  • i was asked to show that the group $(G,+)$ is abelian
    I was asked to show that the group $(G,+)$ is abelian and I already proved that $(0,0)$ is a neutral element of both addition and multiplication and this is the givings $G=\mathbb{Q} \times \mathbb{Z}$ with operations $(a,b)+(c,b)=(a+c , b+d)$ and $(a,b)(c,d)=(2^d a+c , b+d)$
  • Confused by semantics when showing "inequality"
    I believe I've disproved this statement: If $X$ and $Y$ are subspaces of a vector space $V$ and $X \oplus Y \cong V$ (where $\oplus$ denotes the direct sum), then is it true that $V$ is the internal direct sum of $X$ and $Y$? In disproving this statement, I found an example of subspaces $X,Y \subset V$ that share a basis element and $V \cong X \oplus Y$ […]
  • Coordinate ring of $V \subset \mathbb{A}^n$ variety
    Let $V \subset \mathbb{A}^n$ be a nonempty variety and $I(V)$ ideal of V in $K[x_1,...,x_2]$. My question is straightforward: how can I see an element of the quotient $\frac{K[x_1,...,x_2]}{I(V)}$? I'm not referring to $\Gamma(V)=\frac{K[x_1,...,x_2]}{I(V)}$ identified with the subring of $F(V,K)$ consisting of all polynomial functions on $V$. I would l […]
  • When does each trangular diagram commutes imply the entire diagram commutes?
    This question is motivated by a discussion in "A Companion to Lang’s Algebra" by George M. Bergman. Suppose that there is a diagram of say groups as follows: Suppose that there is a group E in the centre of the graph with arrows $e_1,e_2,e_3,e_4$ pointing to $A, B, C, D$ respectively. Then we have a diagram with four triangles. Now even though each […]
  • Exams exercise involving the permutation group $S_5$
    Exercise : (A.1) Show that there exists an element of order $6$ in the group of permutations $S_5$. (A.2) Show that there does not exist an element of order $8$ in the group of permutations $S_5$. (A.3) Find all the possible orders of the elements of the group $S_5$. Attempt : We know that the group $S_5$ is the group of permutations of $5$ objects. If I sta […]

Final Grades

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Most likely they will not be “rolled-in” until Monday.

I wish you have a good break and all the best for your next semester.

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Last minute questions

If you have any last minute questions, I will be in my office from 1 to 2 today.

Last Homework Assignment- Due Monday 12/8

Section 6.8,

Problem 4 (see definitions in section 6.1), 9(a), 10(a),

Section 6.11

Problems 2, 3, 7(a)-(d)

Section 6.12

Problems 1, 3, 14

Section 6.13

Problems 5, 8, 9,

Section 13.2

Problem 6

Section 13.3

Problems 4, 5

Presentation Problem for Monday 12/8

Section 13.2, Problem 4.

Presentation Problem for Wednesday 12/3

Problem based on Section 6.13, Problem 3:

Let G be a group and for each x in G, let T_x be the inner automorphism T_x(a)=x^(-1) a x.

Define a function phi: G->Inner automorphisms of G by phi(x)=T_x.

Prove that the kernel of phi is the center of G (see definition in pg. 144) , and that phi is onto.

Then use the First Isomorphism Theorem to prove that G/Z is isomorphic to the inner automorphisms of G, where Z is the center of G.

Course Evaluations

All students are encouraged to complete their course evaluations. Go to http://www.udmercy.edu/evaluate
Log in with your TitanConnect login credentials.
Complete the course evaluations.
Thank you.

Presentation Problem for Monday 12/1

Section 6.12,
Problem 2.