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Consider $A_{n}$ (to be the subgroup of Upper triangular matrices whose diagonal entries can be any non zero elements from $F_{p}$.) in $Gl_{n}(F_{p})$ (n by n matrices with non zero determinant and $(F_{p}$ is finite field with p elements)). Let $V_{n}$ be the vector space of $C$ - valued function on $Gl_{n}(F_{p})$ (i.e these are functions from $Gl_{n}(F_{p})→C$ ) that are invariant with respect to left and right $A_{n}$-translations. We introduce a binary operation on $V_{n}$ as follows. Given two bi invariant functions, define $(f_{1}⋆f_{2})(g)=∣A_{n}∣1 ∑_{x,y∈GL_{n},xy=g}f_{1}(x)f_{2}(y).$ Consider $V_{3}$ with the convolution product. Denote the function which vanishes outside $A_{3}⊂$ $GL_{3}(F_{p})$ and equals 1 otherwise by $L_{e}$. Let $P_{1}$ (resp. $P_{2}$ ) be subgroup in $GL_{3}(F_{p})$ given by $a_{31}=$ $a_{32}=0($ resp,$a_{21}=a_{31}=0)$. Denote the function which vanishes outside $P_{1}⊂GL_{3}$ and equals 1 otherwise by $L_{1}$, the characteristic function of $P_{2}$ is $L_{2}$. Put $L_{1}=L_{1}−L_{e}$ and $L_{2}=L_{2}−L_{e}$. We are given that the following relations hold. $L_{1}⋆L_{1}=(p−1)L_{1}+pL_{e}L_{2}⋆L_{2}=(p−1)L_{2}+pL_{e}L_{2}⋆L_{1}⋆L_{2}=L_{1}⋆L_{2}⋆L_{1} $ Prove that the elements $T_{1},T_{2}$ generate $V_{3}$

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