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I believe the answer is NO in general. For $n>3$ one obtains a counter-example by taking any maximal torus $T$ corresponding to a partition $\{k, n-k\}$ of $n$ such that $2\leq k \leq n-2$. Clearly $T$ is a $C_{pp}$ group because it doesn't contain any $p$-elements. On the other hand it clearly doesn't preserve an $(n-1)$-dimensional space. Finally it is clear that its order is not divisible by a primitive prime divisor of $q^n-1$ or $q^{n-1}-1$.

Similarly if $n=3$ and $6$ divides $q-1$, then the normalizer of a split torus provides a counter-example. If $6$ does not divide $q-1$, then I'd have to think some more.

If $n=2$, then a counter-example is given by a Sylow $p$-subgroup where $p$ is the characteristic of the field.

Remark: I'm wondering if your definition of $C_{pp}$-group should exclude central elements of $SL_n(q)$. Because, clearly, if the center of $SL_n(q)$ is non-trivial then any $C_{pp}$-subgroup of $SL_n(q)$ must contain no $p$-elements. The counter-examples given above for $n\geq 3$ still work even with the definition you've given, but, still, I wonder if the correct definition of a $C_{pp}$-group should be that the centralizer of any non-trivial $p$-element has form $PZ$ where $P$ is a $p$-group and $Z=Z(SL_n(q))$?

I believe the answer is NO in general. For $n>3$ one obtains a counter-example by taking any maximal torus $T$ corresponding to a partition $\{k, n-k\}$ of $n$ such that $2\leq k \leq n-2$. Clearly $T$ is a $C_{pp}$ group because it doesn't contain any $p$-elements. On the other hand it clearly doesn't preserve an $(n-1)$-dimensional space. Finally it is clear that its order is not divisible by a primitive prime divisor of $q^n-1$ or $q^{n-1}-1$.

Similarly if $n=3$ and $6$ divides $q-1$, then the normalizer of a split torus provides a counter-example. If $6$ does not divide $q-1$, then I'd have to think some more.

If $n=2$, then a counter-example is given by a Sylow $p$-subgroup where $p$ is the characteristic of the field.

Edit: In the comments, the OP has asked for a counter-example where $G$ is simple. For this take $G$ to be $PSL_2(q')$ where $q'$ is a power of $p$. Then $G$ is a simple group and a $C_{pp}$-group. Let $\phi:PSL_2(q')\to GL_n(k)$ be an irreducible representation over $k$, the algebraic closure of the field of order $q'$. This will yield an embedding of $PSL_2(q')$ in $GL_n(q)$ for any $q$ bigger than some constant. Because of irreduciblity $G$ does not lie in $GL_{n-1}(q)$. Now provided we choose $q$ large enough so that $q'^2 < q^{n-1}$, the condition on primitive prime divisors is violated, as required.

I believe the answer is NO in general. For $n>3$ one obtains a counter-example by taking any maximal torus $T$ corresponding to a partition $\{k, n-k\}$ of $n$ such that $2\leq k \leq n-2$. Clearly $T$ is a $C_{pp}$ group because it doesn't contain any $p$-elements. On the other hand it clearly doesn't preserve an $(n-1)$-dimensional space. Finally it is clear that its order is not divisible by a primitive prime divisor of $q^n-1$ or $q^{n-1}-1$.

Similarly if $n=3$ and $3$ divides $q-1$, then the normalizer of a split torus provides a counter-example. If $3$ does not divide $q-1$, then I'd have to think some more.

If $n=2$, then a counter-example is given by a Sylow $p$-subgroup where $p$ is the characteristic of the field.

Remark: I assume, by the way, that your definition of $C_{pp}$-group should exclude central elements of $SL_n(q)$. Because, clearly, if the center of $SL_n(q)$ is non-trivial then any $C_{pp}$-subgroup of $SL_n(q)$ must contain no $p$-elements. The counter-examples given above for $n\geq 3$ still work even with the definition you've given, but, still, I imagine that the correct definition of a $C_{pp}$-group should be that the centralizer of any $p$-element has form $PZ$ where $P$ is a $p$-group and $Z=Z(SL_n(q))$.

I believe the answer is NO in general. For $n>3$ one obtains a counter-example by taking any maximal torus $T$ corresponding to a partition $\{k, n-k\}$ of $n$ such that $2\leq k \leq n-2$. Clearly $T$ is a $C_{pp}$ group because it doesn't contain any $p$-elements. On the other hand it clearly doesn't preserve an $(n-1)$-dimensional space. Finally it is clear that its order is not divisible by a primitive prime divisor of $q^n-1$ or $q^{n-1}-1$.

Similarly if $n=3$ and $6$ divides $q-1$, then the normalizer of a split torus provides a counter-example. If $6$ does not divide $q-1$, then I'd have to think some more.

If $n=2$, then a counter-example is given by a Sylow $p$-subgroup where $p$ is the characteristic of the field.

Remark: I'm wondering if your definition of $C_{pp}$-group should exclude central elements of $SL_n(q)$. Because, clearly, if the center of $SL_n(q)$ is non-trivial then any $C_{pp}$-subgroup of $SL_n(q)$ must contain no $p$-elements. The counter-examples given above for $n\geq 3$ still work even with the definition you've given, but, still, I wonder if the correct definition of a $C_{pp}$-group should be that the centralizer of any non-trivial $p$-element has form $PZ$ where $P$ is a $p$-group and $Z=Z(SL_n(q))$?

I believe the answer is NO for $n>3$. For a counter-example take any maximal torus $T$ corresponding to a partition $\{k, n-k\}$ of $n$ such that $2\leq k \leq n-2$. Clearly $T$ is a $C_{pp}$ group because it doesn't contain any $p$-elements. On the other hand it clearly doesn't preserve an $(n-1)$-dimensional space. Finally it is clear that its order is not divisible by a primitive prime divisor of $q^n-1$ or $q^{n-1}-1$.

Similarly if $n=3$ and the characteristic of the field is $>3$, then the normalizer of a split torus provides a counter-example. For characteristics $2$ and $3$, I'd have to think some more.

If $n=2$, then a counter-example is given by a Sylow $p$-subgroup where $p$ is the characteristic of the field.

I believe the answer is NO in general. For $n>3$ one obtains a counter-example by taking any maximal torus $T$ corresponding to a partition $\{k, n-k\}$ of $n$ such that $2\leq k \leq n-2$. Clearly $T$ is a $C_{pp}$ group because it doesn't contain any $p$-elements. On the other hand it clearly doesn't preserve an $(n-1)$-dimensional space. Finally it is clear that its order is not divisible by a primitive prime divisor of $q^n-1$ or $q^{n-1}-1$.

Similarly if $n=3$ and $3$ divides $q-1$, then the normalizer of a split torus provides a counter-example. If $3$ does not divide $q-1$, then I'd have to think some more.

If $n=2$, then a counter-example is given by a Sylow $p$-subgroup where $p$ is the characteristic of the field.

Remark: I assume, by the way, that your definition of $C_{pp}$-group should exclude central elements of $SL_n(q)$. Because, clearly, if the center of $SL_n(q)$ is non-trivial then any $C_{pp}$-subgroup of $SL_n(q)$ must contain no $p$-elements. The counter-examples given above for $n\geq 3$ still work even with the definition you've given, but, still, I imagine that the correct definition of a $C_{pp}$-group should be that the centralizer of any $p$-element has form $PZ$ where $P$ is a $p$-group and $Z=Z(SL_n(q))$.