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Session 9 Part A Part B Part C Part D Part E Homework
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Session 9 Materials:



Solutions for Session 9, Part H

See solutions for Problems: H1 | H2 | H3 | H4 | H5 | H6| H7| H8| H9| H10| H11

Problem H1



Step 1:


Step 2:

YZYYYZ (erase ZZ)

Step 3:

YZZ (erase YYY)

Step 4:

Y (erase ZZ)



Step 1:


Step 2:

YZZYZY (erase YYY)

Step 3:

YYZY (erase ZZ)

Step 4:

YYYZ (commute last ZY)

Step 5:

Z (erase YYY)



Step 1:

YZYZYZYZYZYZYZYZZZYZYZYYZY (better think more systematically)

Step 2:

YYYYYYYYYYYYYZZZZZZZZZZZZZ (commute all Ys first, Zs last)

Step 3:

YZ (erase 12 Ys by threes, 12 Zs by twos)

<< back to Problem H1


Problem H2

The elements of the YZ group are E, Y, YY, Z, YZ, and YYZ.

<< back to Problem H2


Problem H3


E * YZ = YZ


YZ * YY = Z


Z * YZ = Y

Note that in all of these, any three occurrences of Y can be removed, as can any two occurrences of Z. Since the commutative law exists for this group, order is not important.

<< back to Problem H3


Problem H4


YZ * YYZ = E


Z * Y = YZ


YY * YZ = Z

<< back to Problem H4


Problem H5

Y3 means the same as Y * Y * Y, which is YYY, which is the same as E. The same is true of Z2, which is identical to Z * Z.

<< back to Problem H5


Problem H6

Since Y3 is identical to E, Y4 will be identical to Y, Y5 = YY, Y6 = E, etc.

<< back to Problem H6


Problem H7

Powers of each element:


E: E only


Y: Y1 = Y, Y2 = YY, Y3 = E


YY: YY1 = YY, YY2 = Y, Y3 = E


Z: Z1 = Z, Z2 = E


YZ: YZ1 = YZ, YZ2 = YY, YZ3 = Z, YZ4 = Y, YZ5 = YYZ, YZ6 = E


YYZ: YYZ1 = YYZ, YYZ2 = Y, YYZ3 = Z, YYZ4 = YY, YYZ5 = YZ, YYZ6 = E

<< back to Problem H7


Problem H8


Y1,000 = Y1, since 1,000 = 1 (mod 3), and the powers of Y repeat every three powers.


(YZ)1,001 = (YZ)5, since 1,001 = 5 (mod 6), and the powers of YZ repeat every six powers. According to the list of powers of YZ, (YZ)5 = YYZ. Another way to do this is to imagine a line of 1,001 Ys and 1,001 Zs, and decide what would be left after all the cancellation.

<< back to Problem H8


Problem H9

Here is the completed table:

yz table

Note that every element appears exactly once in each row, and once in each column.

<< back to Problem H9


Problem H10

The element E works this way, since E * A = A for any element A in the table, just like 1 * N = N for any number N.

<< back to Problem H10


Problem H11

The reciprocals can be found by finding E within the row and column of each element. Here are the reciprocals, in pairs:


E and E


Y and YY


Z and Z


YZ and YYZ

<< back to Problem H11


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