How do you find the subgroups of Z12?
Solution. (a) Because Z12 is cyclic and every subgroup of a cyclic group is cyclic, it suffices to list all of the cyclic subgroups of Z12: 〈0〉 = {0} 〈1〉 = Z12 〈2〉 = {0,2,4,6,8,10} 〈3〉 = {0,3,6,9} 〈4〉 = {0,4,8} 〈5〉 = {0,5,10,3,8,1,6,11,4,9,2,7} = Z12 〈6〉 = {0,6}.
What are all the subgroups of Z12?
Z12 is cyclic, so the subgroups are cyclic and are in one-to-one correspon- dence with the divisors of 12. Thus, the subgroups are: H1 = 〈0〉 = {0} H2 = 〈1〉 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} H3 = 〈2〉 = {0, 2, 4, 6, 8, 10} H4 = 〈3〉 = {0, 3, 6, 9} H5 = 〈4〉 = {0, 4, 8} H6 = 〈6〉 = {0, 6}.
What is the order of 9 in Z12?
All other elements other than 0 have order 9. (c) In the group Z12, the elements 1, 5, 7, 11 have order 12. The elements 2, 10 have order six. The elements 3, 9 have order four.
How do you describe a subgroup?
A subgroup of a group G is a subset of G that forms a group with the same law of composition. For example, the even numbers form a subgroup of the group of integers with group law of addition.
How many different subgroups are there for z_12?
You should find 6 subgroups. Hint: If a subgroup contains an element n, then it also contains n+n,n+n+n,…
What are the subgroups of Z8?
(Subgroups of a finite cyclic group) List the elements of the subgroups generated by elements of Z8. 〈0〉 = {0}, 〈2〉 = 〈6〉 = {0, 2, 4, 6}, 〈4〉 = {0, 4}, 〈1〉 = 〈3〉 = 〈5〉 = 〈7〉 = {0, 1, 2, 3, 4, 5, 6, 7}.
How many subgroups are there for the group Z12?
You should find 6 subgroups.
What is the order of element 6 8 in the Quotient group Z48 8?
Elements of (8) ⊂ Z48 are {0,8,16,24,32,40} and |(8)| = 6. Therefore the order of the factor group is: |Z48/(8)| = |Z48|/|(8)| = 48/6 = 8.
How many subgroups does Z20 have?
(e) Draw the subgroup lattice of Z20 [Note: 20 = 22 · 5]. We know that there is exactly one subgroup per divisor of 20. These subgroups are arranged ac- cording to divisibility, so to draw a subgroup lat- tice we should first draw a divisibility lattice for the divisors of 20.
What is the subgroup of Z?
The proper cyclic subgroups of Z are: the trivial subgroup {0} = 〈0〉 and, for any integer m ≥ 2, the group mZ = 〈m〉 = 〈−m〉. These are all subgroups of Z. Theorem Every subgroup of a cyclic group is cyclic as well. Proof: Suppose that G is a cyclic group and H is a subgroup of G.
What is s sub 3?
It is the symmetric group on a set of three elements, viz., the group of all permutations of a three-element set. In particular, it is a symmetric group of prime degree and symmetric group of prime power degree.
