Question

Find a cyclic subgroup of a8 that has order 4. find a noncyclic subgroup of a8 that has order 4.

Answer 1

A cyclic subgroup of A8 with order 4 is generated by the cycle (1234), resulting in the subgroup {e, (1234), (13)(24), (1432)}. A noncyclic subgroup of A8 with order 4 can be formed by permutations that are products of two disjoint 2-cycles, for example, {e, (12)(34), (56)(78), (12)(56)}.

Step-by-step explanation:

To find a cyclic subgroup of A8 (the alternating group on 8 elements) that has order 4, we can use the cycle notation. One simple cycle of order 4 in A8 is (1234). This cycle represents a permutation where 1 goes to 2, 2 goes to 3, 3 goes to 4, and 4 goes back to 1, with all other elements remaining fixed. The subgroup generated by this cycle, denoted by {e, (1234), (13)(24), (1432)}, is cyclic and has order 4, where e represents the identity permutation.

To find a noncyclic subgroup of A8 with order 4, we can consider a group consisting of permutations that are the product of two disjoint 2-cycles, which do not generate a cyclic group. An example of such a subgroup is {e, (12)(34), (56)(78), (12)(56)}. This subgroup has the required four elements but is not cyclic because there is no single element in the subgroup that can be raised to successive powers to generate all the other elements.

Answer 2

A cyclic subgroup of A8 with order 4 can be formed by the permutation (1234). A noncyclic subgroup of A8 with order 4 can include permutations like {(12)(34), (56)(78), (12)(56), (34)(78)}, consisting of products of disjoint transpositions.

Step-by-step explanation:

To find a cyclic subgroup of A8 (the alternating group on 8 elements) with order 4, we can consider a 4-cycle permutation. For instance, let's take the permutation (1234), which is in A8 since it is an even permutation. The cyclic subgroup generated by (1234) is {e, (1234), (13)(24), (1432)}, where e represents the identity permutation. This set forms a subgroup because it is closed under the group operation, contains the identity element, contains the inverse of each element, and is associative, as all subgroups must be.

To find a noncyclic subgroup of A8 with order 4, consider the permutations that are products of disjoint transpositions, like {(12)(34), (56)(78), (12)(56), (34)(78)}. This set has order 4 but is not cyclic as no single element raised to successive powers will generate the entire group. This subset forms a subgroup because these elements are even permutations, so remain in A8, and they satisfy the subgroup criteria outlined above.