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**tttcomrader** Suppose that $\displaystyle G = C_6 \times C_6 \times C_6$, $\displaystyle C_6$ generated by an element r. Let $\displaystyle H = \{ (a,a,a) \in G : a \in C_6 \}$, $\displaystyle J_1 = \{ (a,1,1) \in G : a \in C_6 \}$, $\displaystyle J_2 = \{ (1,a,1) \in G : a \in C_6 \}$, $\displaystyle J_3 = \{ (1,1,a) \in G : a \in C_6 \}$

Prove that $\displaystyle G=H \oplus I \oplus J_i \ \ \ \ i=1,2 $, but $\displaystyle G \neq H \oplus I \oplus J_3 $

Proof.

Suppose that $\displaystyle g \in G$, then $\displaystyle g = abc \ \ \ a,b,c \in C_6 $, I want to get g to somehow be in the form [tex]