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HallsofIvy

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Do you know what "equivalence relations" and "equivalence classes" are?

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My favorite quotient groups is the numbers on a face clock.

The numbers ...-11,1,13,25... are an equivalence class, with representative 1.

Thus, the infinite set of integers ...-1,0,1,2,3,... is partitioned into 12 sets, or cosets. Since the theorem of quotient groups holds, these 12 representatives have a group structure inherited from the group structure of the integers. The operation in question is addition. So, let's take two representatives in the quotient group, say 8 and 7, then 8+7 is 15, however we might prefer to represent 15 with another integer in it's class, say 3. This is how we define the inherited operation of addition in the quotient group.

Let's see how this corresponds with the steps used in general quotient groups. What steps do we use to get from the integers to the group with elements {1,2,...,12}, in other words, how do we construct the latter group. Let Z represent the group of integers, let 12Z represent the subgroup in Z generated by 12, ie 12Z={...-12,0,12,24,36,...}. Then the numbers on the clock are constructed via the symbolism

Z/(12Z).

Oops, gotta go, I'll let others continue this. Please let us know where you're at on this.

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