6. List every generator for the subgroup of order 8 in Z32.
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- 4. List all the elements of the subgroupin the group under addition, and state its order.Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6 ],[ 9 ] } be subgroups of the abelian group 12 under addition. Find H1+H2 and determine if the sum is direct.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?
- 1. Consider , the groups of units in under multiplication. For each of the following subgroups in , partition into left cosets of , and state the index of in a. b.Show that An has index 2 in Sn, and thereby conclude that An is always a normal subgroup of Sn.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 ] .