Let G be a transitive permutation group on the finite set A. A block is a...

   

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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 o E G either o (B) = B or o (B) nB = (here o (B) is the set {o (b) | b e B)). (a) Prove that if B is a block containing the element a of A, then the set GB defined by GB = {o e Go (B) = B) is a subgroup of G containing Ga. (b) Show that if B is a block and 0 (B), 02(B),..., on (B) are all the distinct images of B under the elements of G, then these form a partition 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 S4 is primitive on A = {1, 2, 3, 4). Show that Dg is not primitive as a permutation group on the four vertices of a square. (d) Prove that the transitive group G is primitive on A if and only if for each a A, the only subgroups of G containing Ga are Ga and G [Use part (a).] honce 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 o E G either o (B) = B or o (B) nB = (here o (B) is the set {o (b) | b e B)). (a) Prove that if B is a block containing the element a of A, then the set GB defined by GB = {o e Go (B) = B) is a subgroup of G containing Ga. (b) Show that if B is a block and 0 (B), 02(B),..., on (B) are all the distinct images of B under the elements of G, then these form a partition 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 S4 is primitive on A = {1, 2, 3, 4). Show that Dg is not primitive as a permutation group on the four vertices of a square. (d) Prove that the transitive group G is primitive on A if and only if for each a A, the only subgroups of G containing Ga are Ga and G [Use part (a).] honce