## Groups acting on themselves by left multiplication-Cayley’s theorem[Problems in the section 4.2 of ”Dummit and Foote: Abstract Algebra”]

These are  problems,  Its solutions will coming soon! Pdf file 42_problems

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Solutions 42_solutions

## Group actions and permutation representations[Problems in the section 4.1 of ”Dummit and Foote: Abtract Algebra”]

Let $G$ be a group and let $A$ be a nonempty set.

1. Let $G$ act on the set $A$. Prove that if $a,b\in A$ and $b=g\cdot a$ for some $g\in G$, then $G_b=gG_ag^{-1}$($G_a$ is the stabilizer of $a$). Deduce that if $G$ acts transitively on $A$ then the kernel of the action is $\cap_{g\in G}gG_ag^{-1}$.

2. Let $G$ be a permutation group on the set $A$(i.e., $G\leq S_A$), let $\delta \in G$ and let $a\in A$. Prove that $\delta G_a\delta^{-1}=G_{\delta(a)}$. Deduce that if $G$ acts transitively on $A$ then $\cap_{\delta \in G}\delta G_a\delta^{-1}=1.$

3. Assume that $G$ is an abelian, transitive supgroup of $S_A$. Show that $\delta (a)\not =a\forall \delta\in G-\{1\}\forall a\in A$. Deduce that $|G|=|A|$[Use the preceding exercise.]

4. Let $S_3$ act on the set $\Omega$ of ordered pairs: $\{(i,j)|1\leq i,j\leq 3\}$ by $\delta ((i,j))=(\delta (i),\delta (j))$. Find the orbits of $S_3$ on $\Omega$. For each $\delta \in S_3$ find the cycle decomposition of $\delta$ under this action (i.e., find its cycle decomposition when $\delta$ is considered as an element of $S_9$– first fix a labelling of these nine ordered pairs). For each orbit $\mathcal{O}$ of $S_3$ acting on these nine points pick some $a\in\mathcal{O}$ and find the stabilizer of $a$ in $S_3$.

5. For each parts (a) and (b) repeat the preceding exercise but with $S_3$ action on the specified set:

(a)The set of $27$ triples $\{(i,j,k)|1\leq i,j,k\leq 3\}$

(b)The set $\mathcal{P}(\{1,2,3\})-\{\emptyset\}$ of all $7$ nonempty subsets of $\{1,2,3\}$.

6. Let $R$ be the set of all polynomials with integer coefficients in the independent variables $x_1,x_2,x_3,x_4$ and $S_4$ act on $R$ by permuting the indices of the four variables:$\sigma\cdot p(x_1,x_2,x_3,x_4)=p(x_{\sigma (1)},x_{\sigma (2)},x_{\sigma (3)},x_{\sigma (4)})$ for all $\sigma \in S_4$ and $p\in R$.

a)Find the polynomials in the orbit of $S_4$ on $R$ containing $x_1+x_2$;

b)Find the polynomials in the orbit of $S_4$ on $R$ containing $x_1x_2+x_3x_4$;

c)Find the polynomials in the orbit of $S_4$ on $R$ containing $(x_1+x_2)(x_3+x_4)$.

7. Let $G$ be a transitive permutation group on the finite set $A$. A block is a nonempty subset $B$ of $A$ such that for all $\sigma \in G$ either $\sigma (B)=B$ or $\sigma (B)\cap B=\emptyset$.

a)Prove that if $B$ is a block containing the element $a$ of $A$ then $G_B:=\{\sigma \in G|\sigma (B)=B\}$ is a subgroup of $G$ containing $G_a$;

b)Show that if $B$ is a block and $\sigma_1(B),\cdots,\sigma_n(B)$ are all dinstinct images of $B$ under the elements of $G$ then these form a partion of $A$;

c)A transitive group $G$ on a set $A$ is said to be primitive if the only blocks in $A$ are the trivial ones: the sets of size $1$ and $A$ itself. Show that $S_4$ is primitive on $A=\{1,2,3,4\}$. Show that $D_8$ is not primitive as a permutation group on the four vertices of a square;

d)Prove that the transitive group is primitive of $A$ iff for each $a\in A$, the only subgroups of $G$ containing $G_a$ are $G_a$ and $G$.

8. A transitive permutation group $G$ on a set $A$ is called doubly transitive if for any (hence all) $a\in A$ the subgroup $G_a$ is transitive on the set $A-\{a\}$.

a)Prove that $S_n$ is doubly transitive on $\{1,2,\cdots,n\}$ for all $n>1$;

b)Prove that a doubly transitive group is primitive. Deduce that $D_8$ is not doubly transitive in its action on the $4$ vertices of a square.

9. Assume $G$ acts transitively on the finite set $A$ and let $H$ be a normal subgroup of $G$. Let $\mathcal{O}_1,\mathcal{O}_2,\cdots,\mathcal{O}_r$ be the distinct orbits of $H$ on $A$.

a)Prove that $G$ permutes the sets $\mathcal{O}_i$. Prove that $G$ is transitive on $\{\mathcal{O}_i\}$. Deduce that all orbits of $H$ on $A$ have the same cardinality;

b)Prove that if $a\in\mathcal{O}_1$ then $|\mathcal{O}_1|=|H:H\cap G_a|$ and $r=|G:HG_a|$.

10. Let $H$ and $K$ be subgroups of the group $G$. For each $x\in G$ define the $HK$ double coset of $x$ in $G$ to be the set $HxK=\{hxk|h\in H, k\in K\}.$

a)Prove that $HxK$ is the union of the left cosets $x_iK$, where $\{x_iK\}$ is the orbit containing $xK$ of $H$ acting by left multiplication on the set of left cosets of $K$;

b)Prove that $HxK$ is the union of right cosets of $H$;

c)Prove that $HxK$ and $HyK$ are either the same set or are disjoint for all $x,y\in G$. Show that the set of $HK$ double cosets partitions $G$;

d)Prove that $|HxK|=|K|\cdot |H:H\cap xKx^{-1}|$;

e)Prove that $|HxK|=|H|\cdot |K:K\cap x^{-1}Hx|$.

P.S. These problems are from ”Dummit and Foote, Abstract Algebra”. Solutions will coming soon! 😀

Pdf file: 41_problems

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Solutions:41_solutions

## Kiểm tra Đại số hiện đại, ngày 12-11-2008 (thày Phùng Hồ Hải)

Bài 1. Cho V là không gian các đa thức có bậc nhỏ hơn n với hệ số phức, ở đây n>1 là một số nguyên. Xét toán tử đạo hàm $d:V\to V$. Tồn tại hay không một toán tử $g:V\to V$ là đa thức theo d và thoả mãn V là tổng trực tiếp của Im(d) và Im(g).

Bài 2. V là không gian véc tơ hữu hạn chiều bất kỳ trên $\mathbb{C}$$f:V\to V$ là một ánh xạ tuyến tính.

a)Chứng minh rằng tồn tại các ánh xạ tuyến tính $f_s,f_n:V\to V$ thoả mãn $f_s$ chéo hoá được, $f_n$ luỹ linh, $f_s,f_n$ là các đa thức theo $f$$f=f_s+f_n$.

b) Chứng minh rằng cặp ánh xạ trên là duy nhất.

Bài 3. Cho k là một trường có đặc số 0. Xét đại số nhóm $k[S_n]$ với cơ sở $e_{\sigma},\sigma\in S_n$. Chứng minh rằng $x^2=x$ với $x=\dfrac{1}{n!}\sum_{\sigma\in S_n}\epsilon (\sigma)e_{\sigma}$. ($\epsilon (\sigma)$ là dấu của hoán vị $\sigma$)

Bài 4. Cho $f:M\to M$ là một đồng cấu modun thoả mãn $f^2=f$. Chứng minh rằng tồn tại modun con N của M để M là tổng trực tiếp của Im(f) và N.

Bài 5. Bằng định nghĩa tích tensor chứng minh rằng $\mathbb{Z}/(m,n)\mathbb{Z}$ đẳng cấu với tích tensor của hai nhóm abel $\mathbb{Z}/m\mathbb{Z}$$\mathbb{Z}/n\mathbb{Z}$.

## Bài tập Đại số hiện đại, 30-10-2008(thày Phùng Hồ Hải)

Bài 1. Tính $Hom_{\mathbb{Z}}(\mathbb{Z},\mathbb{Q}); Hom_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z}); Hom_{\mathbb{Z}}(\mathbb{Z}_n,\mathbb{Z}_m);Hom_{\mathbb{Z}}(\mathbb{Z}_n,\mathbb{Q})$$Hom_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z}_n)$.

Bài 2. Cho $A$ là nhóm Abel hữu hạn sinh. Chứng minh rằng $Hom_{\mathbb{Z}}(A,\mathbb{Q}/\mathbb{Z})\not =0$. Khi nào thì có đơn cấu từ $A$ đến $\mathbb{Q}/\mathbb{Z}$?

Bài 3. Cho vành R và M là một R-modun đơn. Chứng minh rằng $End_R(M)$ là một thể.

Bài 4. V là K-không gian véc tơ hữu hạn chiều, $R=End_K(V)$. Coi V là modun trên R. Tính $End_R(V)$.

## Bài giảng về đại số Lie và nhóm Lie

Introduction to Lie Algebras and Lie Groups

Fall 2008

International Master Class

Institute of Mathematics

Vietnam Academy of Science and Technology

This course will cover the basic theory of Lie groups and Lie algebras. The prequisites include knowledge of linear algebra and group theory as covered by Algebra courses and basic notions of differential geometry (manifolds, vector fields,… etc).

 TIME and PLACE 13:30 – 16:00, Monday, Wednesday and Thursday at Lecture hall 301A, Building A5. The first lecture will be held on Wednesday, October 29, 2008. INSTRUCTOR Professor Pierre Cartier, IHES The best way to contact Professor P. Cartier is during the lecture or at his office (room 110, building A5) CONTENTS Introduction: Global and infinitesimal symmetris Lie algebras: Basic definitions, enveloping algebra, Hopf lgebras, classical Lie algebras, Cartan subalgebras (roots and weights) Lie groups: Classical Lie groups, Lie algebra of a Lie group, algebraic groups, maximal torus and Bruhat decomposition Basic results about linear representations A glimpse into modern developments: Quantum groups , Lie groupoids TEXTBOOKS A. Kirillov Jr., Introduction to Lie Groups and Lie Algebras, Cambridge University Press, 2002 N. Bourbaki, Lie groups and Lie algebras Chapter 1-3 ISBN 3-540-64242-0, Chapters 4-6 ISBN 3-540-42650-7, Chapters 7-9 ISBN 3-540-43405-4 J. P. Serre, Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University, LNM 1500, Springer R. Carter et al., Lectures on Lie Groups and Lie Algebras, LMS Student Texts Series, 1995 J. E. Humphreys, Introduction to Lie Algebras and Representaion theory, Springer 1978 Note: Almost all of these textbooks are available at the library of the Institute of Mathematics. Some of them are available electronically also. FINAL EXAM The final exam will be posted here.

P.S. Cái này mình copy trên trang của Viện Toán, bác nào rảnh thì đi nghe nhá!