Overview
The deifning equation of the Lorentz transformation {% \Lambda %} is given as
{% \Lambda ^T \eta \Lambda = \eta %}
where {% \eta %} is the Lorentz metric. Taking the
determinant
of both sides of the equation, we get
{% det(\Lambda) det(\eta) det(\Lambda) = det(\eta) \rightarrow det(\Lambda) = \pm 1 %}
Group Structure
Lorentz transformations form a lie group. In particular, two different Lorentz transformations applied one after the other, results in a new Lorentz transformation. This can be verified by plugging into the defing equation.
{% \Lambda_2^T \Lambda_1 ^T \eta \Lambda_1 \Lambda_2 = \Lambda_2^T \eta \Lambda_2 = \eta %}
That is, {% \Lambda_1 \Lambda_2 %} forms a Lorentz transformation.