Chứng minh Định lý Cauchy của James McKay


Hôm nay tôi sẽ giới thiệu với các bạn một chứng minh của Định lý sau

Định lý Cauchy. Cho G là một nhóm hữu hạn và p là một ước nguyên tố của |G|. Khi đó G có ít nhất một phần tử cấp p.

Chứng minh.

Xét tập \mathcal{S}=\{(x_1,x_2,\cdots, x_p)\in G^p|x_1x_2\cdots x_p=1\}. Bởi vì x_p xác định duy nhất khi ta đã biết x_1,\cdots, x_{p-1} nên số phần tử của \mathcal{S} bằng |G|^{p-1}. Trong \mathcal{S} xét quan hệ sau: x\,\,\,R\,\,\,y nếu x thu được từ y bởi phép hoán vị vòng. Dễ thấy đây là một quan hệ tương đương, và một lớp theo quan hệ này có một phần tử khi và chỉ khi nó chứa (x,\cdots, x) với x^p=1. Cũng thấy luôn rằng vì p là số nguyên tố nên một lớp tương đương chỉ có thể có 1 hoặc p phần tử. Gọi k là số lớp có 1 phần tử còn q là số lớp có p phần tử, thế thì ta sẽ có |G|^{p-1}=k+pq, từ đây ta có p|k, nói riêng k>1. Như vậy ngoài lớp chứa (1,1,…,1) còn có những  lớp khác cũng gồm một phần tử, giả sử một trong các lớp này chứa (x,x,…,x) thì x là phần tử có bậc p. Định lý được chứng minh.

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Theo AMM

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