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Abstract Algebra

Exercise 4.5.39 ($U_n(\mathbb{F}_p)$ is a Sylow $p$-subgroup of $\mathrm{GL}_n(\mathbb{F}_p)$)

Show that the subgroup of strictly upper triangular matrices in GL n ( 𝔽 p ) (cf. Execise 17, Section 2.1) is a Sylow p -subgroup of this finite group. [Use the order formula in Section 1.4 to find the order of a Sylow p -subgroup of GL n ( 𝔽 p ) .]

Answers

Proof. By Exercise 2.1.17,

U n ( 𝔽 p ) = { ( a 𝑖𝑗 ) ( i , j ) ∈ [ [ 1 , n ] ] 2 ∈ GL n ( F ) ∣ ∀ ⁡ ( i , j ) ∈ [ [ 1 , n ] ] 2 , i > j ⇒ a 𝑖𝑗 = 0  and  ∀ ⁡ i ∈ [ [ 1 , n ] ] , a 𝑖𝑖 = 1 }

is a subgroup of GL n ( 𝔽 p ) .

We know (see Section 1.4) that

| GL n ( 𝔽 p ) | = ∏ i = 0 n − 1 ( p n − p i ) = p n ( n − 1 ) 2 ∏ i = 1 n ( p i − 1 ) .

Since p ∤ ∏ ⁡ i = 1 n ( p i − 1 ) , the p -Sylow subgroups of GL n ( 𝔽 p ) are the subgroups of order p n ( n − 1 ) 2 .

Moreover, to build an element of U n ( 𝔽 p ) , it suffices to choose the elements a i , j ∈ p with i < j . There are n ( n − 1 ) 2 ordered pairs ( i , j ) such that i < j , therefore

| U n ( 𝔽 p ) | = p n ( n − 1 ) 2 .

This proves that U n ( 𝔽 p ) is a Sylow p -subgroup of GL n ( 𝔽 p ) . □

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2026-04-01 10:16
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