Finding order and cosets of subgroup

Question

For:

Let $P$ be the pairs $(a,b)$ where $a \in \Bbb Z_4$, and $b \in \Bbb Z_2$

An operation, $*$, is defined by: $$(a,b)*(c,d)=(a+c \pmod 4, b+d \pmod 2)$$ for all $(a,c),(b,d)\in P$

H = ⟨(1,1)⟩ is the cyclical subgroup generated by (1,1)

How do I find the order of $H$ and the cosets of $H$?

Answer 1

Hint: Here $1_4$ is a generator of $(\Bbb Z_4, +_4)$ and $1_2$ is a generator of $(\Bbb Z_2, +_2)$.

For a further hint, click, hover over, or tap the box below.

Since $\Bbb Z_4\times \Bbb Z_2$ has only eight elements, it is feasible & practical to simply compute all the powers of $(1_4,1_2)$.

Another hint:

Use Lagrange's Theorem.