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

Polynomials by Victor V. Prasolov


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