Find the partition of Ze into cosets of the subgroup H = {0,3}.
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- Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=4022. If and are both normal subgroups of , prove that is a normal subgroup of .12. Find all normal subgroups of the quaternion group.
- Find two groups of order 6 that are not isomorphic.Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?For each of the following subgroups H of the addition groups Z18, find the distinct left cosets of H in Z18, partition Z18 into left cosets of H, and state the index [ Z18:H ] of H in Z18. H= [ 8 ] .