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Ch 11.6: Direct Products of Groups The following questions work in the product space [Z7, +7] × [Z4, ×7]. 6 What is the identity element in this group? 7 Determine the powers (2, 2)$^2$, (2, 2)$^4$, (2, 2)$^8$, (2, 2)$^{16}$. Note: Use the fact that (2, 2)$^4$ = (2, 2)$^2$ × (2, 2)$^2$, and proceed similarly for the other powers. 8 Use your answers to the previous question to determine (2, 2)$^{21}$ and (2, 2)$^{22}$. Hint: (2, 2)$^{21}$ = (2, 2)$^{16+4+1}$, 9 Use your answer to the previous question to determine (2, 2)$^{-1}$.

          Ch 11.6: Direct Products of Groups
The following questions work in the product space [Z7, +7] × [Z4, ×7].
6 What is the identity element in this group?
7 Determine the powers (2, 2)$^2$, (2, 2)$^4$, (2, 2)$^8$, (2, 2)$^{16}$.
Note: Use the fact that (2, 2)$^4$ = (2, 2)$^2$ × (2, 2)$^2$, and proceed similarly for the other powers.
8 Use your answers to the previous question to determine (2, 2)$^{21}$ and (2, 2)$^{22}$.
Hint: (2, 2)$^{21}$ = (2, 2)$^{16+4+1}$,
9 Use your answer to the previous question to determine (2, 2)$^{-1}$.

Ch 11.6: Direct Products of Groups
The following questions work in the product space [Z7, +7] × [Z4, ×7].
6 What is the identity element in this group?
7 Determine the powers (2, 2)^2, (2, 2)^4, (2, 2)^8, (2, 2)^16.
Note: Use the fact that (2, 2)^4 = (2, 2)^2 × (2, 2)^2, and proceed similarly for the other powers.
8 Use your answers to the previous question to determine (2, 2)^21 and (2, 2)^22.
Hint: (2, 2)^21 = (2, 2)^16+4+1,
9 Use your answer to the previous question to determine (2, 2)^-1.

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