Find all generators of the subgroup of Z/60Z with order 12.
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Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…
A: 2Z ={ ......... , -8, -6 , -4 , -2 , 0 , 2, 4, 6 , 8, ....}
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Find all generators of the subgroup of Z/60Z with order 12.
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- Find all subgroups of the quaternion group.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.3. Consider the group under addition. List all the elements of the subgroup, 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.9. Find all homomorphic images of the octic group.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 ] .
- 40. Find the commutator subgroup of each of the following groups. a. The quaternion group . b. The symmetric group .Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.Let H1 and H2 be cyclic subgroups of the abelian group G, where H1H2=0. Prove that H1H2 is cyclic if and only if H1 and H2 are relatively prime.