Let be a group and let be a nonempty set.
1. Let act on the set . Prove that if and for some , then ($G_a$ is the stabilizer of ). Deduce that if acts transitively on then the kernel of the action is .
2. Let be a permutation group on the set (i.e., ), let and let . Prove that . Deduce that if acts transitively on then
3. Assume that is an abelian, transitive supgroup of . Show that . Deduce that [Use the preceding exercise.]
4. Let act on the set of ordered pairs: by . Find the orbits of on . For each find the cycle decomposition of under this action (i.e., find its cycle decomposition when is considered as an element of – first fix a labelling of these nine ordered pairs). For each orbit of acting on these nine points pick some and find the stabilizer of in .
5. For each parts (a) and (b) repeat the preceding exercise but with action on the specified set:
(a)The set of triples
(b)The set of all nonempty subsets of .
6. Let be the set of all polynomials with integer coefficients in the independent variables and act on by permuting the indices of the four variables: for all and .
a)Find the polynomials in the orbit of on containing ;
b)Find the polynomials in the orbit of on containing ;
c)Find the polynomials in the orbit of on containing .
7. Let be a transitive permutation group on the finite set . A block is a nonempty subset of such that for all either or .
a)Prove that if is a block containing the element of then is a subgroup of containing ;
b)Show that if is a block and are all dinstinct images of under the elements of then these form a partion of ;
c)A transitive group on a set is said to be primitive if the only blocks in are the trivial ones: the sets of size and itself. Show that is primitive on . Show that is not primitive as a permutation group on the four vertices of a square;
d)Prove that the transitive group is primitive of iff for each , the only subgroups of containing are and .
8. A transitive permutation group on a set is called doubly transitive if for any (hence all) the subgroup is transitive on the set .
a)Prove that is doubly transitive on for all ;
b)Prove that a doubly transitive group is primitive. Deduce that is not doubly transitive in its action on the vertices of a square.
9. Assume acts transitively on the finite set and let be a normal subgroup of . Let be the distinct orbits of on .
a)Prove that permutes the sets . Prove that is transitive on . Deduce that all orbits of on have the same cardinality;
b)Prove that if then and .
10. Let and be subgroups of the group . For each define the double coset of in to be the set
a)Prove that is the union of the left cosets , where is the orbit containing of acting by left multiplication on the set of left cosets of ;
b)Prove that is the union of right cosets of ;
c)Prove that and are either the same set or are disjoint for all . Show that the set of double cosets partitions ;
d)Prove that ;
e)Prove that .
P.S. These problems are from ”Dummit and Foote, Abstract Algebra”. Solutions will coming soon! 😀
Pdf file: 41_problems