Consider the "clock arithmetic" group (Z_15, oplus)a) Using Lagrange`s Theotem, state all possible orders for subgroups of this group.b) List all of the subgroups of (Z_15, oplus)

geduiwelh

geduiwelh

Answered question

2021-01-31

Consider the "clock arithmetic" group (Z15,) a) Using Lagrange`s Theotem, state all possible orders for subgroups of this group. b) List all of the subgroups of (Z15,)

Answer & Explanation

hesgidiauE

hesgidiauE

Skilled2021-02-01Added 106 answers

Solution: Given group G=(Z15,) a) We know that by Lagrange`s theorem order of subgroup divide the order of group. Since O(G)=15 Let H be subgroup of G. Then O(H)|O(G)O(H)|15 Possible order of H are 1, 3, 5, 15 Since G(Z15,) is cycling group. Then every divisor of order of group has subgroup. Then b)

H1={e}

H2=<5>={5,10,0}

H3=<5>{3,6,9,12,0}

H4=<1>={1,2,3,4,5,6,7,8,9,10,11,12,13,14,0}

These are H1,H2,H3andH4 are subgroup of group G=(Z15,)

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