Common gain ,Relation between alpha α beta β gamma γ

Common gain

Current gain or current amplification factor is defined by the ratio of output current to input current when output voltage is kept constant

CB Configuration
$\left. α=\frac{ΔI_{C}}{ΔI_{E}}  \right |_{V_{CB}=constant }$
CE Configuration
$\left. β=\frac{ΔI_{C}}{ΔI_{B}}  \right |_{V_{CE}=constant }$
CC Configuration 

$\left. γ=\frac{ΔI_{E}}{ΔI_{B}}  \right |_{V_{CE}=constant }$

Relation between alpha α beta β gamma γ

We know that 

$I_E=I_B+I_C$→1

dividing it by  $I_C$ gives

$\frac{I_E}{I_C}=\frac{I_B}{I_C}+\frac{I_C}{I_C}$
$\frac{1}{α }=\frac{1}{β }+\frac{1}{1}=\frac{1}{β }+1$
$α =\frac{β}{β+1 }$  

$\frac{1}{β }=\frac{1}{α }-1$ 
$\frac{1}{β }=\frac{1-α}{α }$ 
$β =\frac{α }{1-α}$ 

dividing it by  $I_B$ gives

 $\frac{I_E}{I_B}=\frac{I_B}{I_B}+\frac{I_C}{I_B}$
$γ=1+β$
$β=γ-1$