This group has 8 elements:

$0=(0,0,0)$

$a=(1,0,0)$

$b=(0,1,0)$

$c=(1,1,0)$

$d=(0,0,1)$

$f=(1,0,1)$

$g=(0,1,1)$

$h=(1,1,1)$

A generic element of this set shall be represented as $x=(x_{1}, x_{2}, x_{3})$, where $x_{1}, x_{2}, x_{3} \in \mathbb{Z}_{2}$

**Verification of group:**

Table:

+ | 0 | a | b | c | d | f | g | h |

0 | 0 | a | b | c | d | f | g | h |

a | a | 0 | c | b | f | d | h | g |

b | b | c | 0 | a | g | h | d | f |

c | c | b | a | 0 | h | g | f | d |

d | d | f | g | h | 0 | a | b | c |

f | f | d | h | g | a | 0 | c | b |

g | g | h | d | f | b | c | 0 | a |

h | h | g | f | d | c | b | a | 0 |

Associativity:

$x, y, z \in \mathbb{Z}_{2} ^3$

Then,

$(x_{1}+y_{1})+z_{1}=x_{1}+(y_{1}+z_{1})$,

$(x_{2}+y_{2})+z_{2}=x_{2}+(y_{2}+z_{2})$, and

$(x_{3}+y_{3})+z_{3}=x_{3}+(y_{3}+z_{3})$ by associativity of $+_{2}$ in $\mathbb{Z}_{2}$.

So,

$(x+y)+z=x+(y+z)$]

Therefore, component addition is associative in $\mathbb{Z}_2^3$.

Identy:

As the table shows, 0 is the identity.

Inverse:

As the table shows, each element is its own inverse.

**Order of group:**

$|\mathbb{Z}_2^3|=8$

**Order of elements:**

Because every element is its own inverse,

$|a|=|b|=|c|=|d|=|f|=|g|=|h|=2$

Because 0 is the identity, it generates the trivial subgroup, {0}, so $|0|=1$

**Commutivity**

Let $x,y\in\mathbb{Z}_2^3$ be arbitrary.

$x=(j,k,l), y=(m,n,p)$ where $j,k,l,m,n,p\in\mathbb{z}_2$

$x+y=(j+m,k+n,l+p)$ by definition of component addition

$x+y=(m+j,n+k,p+l)$ by commutivity of addition in $\mathbb{z}_2$

$x+y=(m,n,p)+(j,k,l)$ by definition of component addition

$x+y=y+x$

Therefore, $\mathbb{Z}_2^3$ is abelian

**Generators**

Generating sets:

{a,b,d},{a,b,f},{a,b,g},{a,b,h},{a,c,d},{a,c,f},{a,c,g},{a,c,h},{a,d,g},{a,d,h},{a,f,g},{a,f,h},{b,c,d},{b,c,f},{b,c,g},{b,c,h},{b,d,f},{b,d,h},{b,f,g},{b,g,h},{c,d,f},{c,d,g},{c,f,h},{d,f,g},{d,f,h},{f,g,h}

Any set of three elements such that the plane in $\mathbb{R}^3$ defined by those three elements contains no other elements of $\mathbb{Z}_2^3$, except {c,f,g}. Also, any set of three non-identity elements where two do not sum to the third.

**Subgroups**

###### Order 2:

(all subgroups generated by a single element)

{e,a},{e,b},{e,c},{e,d},{e,f},{e,g},{e,h}

###### Order 4:

(all subgroups generated by two elements)

{e,a,b,c},{e,a,d,f},{e,a,g,h},{e,b,d,g},{e,b,f,h},{e,c,d,h},{e,c,f,g}

**Cosets**

The cosets of {e,a} are {e,a},{b,c},{d,f}, and {g,h}.

These cosets are the edges of the cube that are parellel to {e,a}.