True-or-False Applications of Sylow Theorems

The final part of this section turns Sylow theory into a diagnostic tool for true-or-false questions. Such problems are valuable because they test whether a student understands the exact conclusion of each theorem. In this lesson, we focus in the true-or-false applications of Sylow theorems . The central habit is to check both parts of a claim: existence of a subgroup and normality of that subgroup . A statement may fail because a subgroup exists but is not unique, or because uniqueness gives a subgroup but not a unique element.

DEFINITION

TRUE-OR-FALSE SYLOW TEST

A \textbf{true-or-false Sylow test} is a verification method in which Sylow congruence, divisibility, subgroup order, and normality are checked to prove a statement or construct a counterexample.

This preview helps identify the exact issue in a true-or-false Sylow statement. Choose a statement from the lesson and read whether the key distinction is existence, uniqueness, normality, element count, or isomorphism . The goal is to locate the precise theorem being used and the precise conclusion it allows. This prevents the common mistake of turning existence into normality or uniqueness of a subgroup into uniqueness of an element.

Dynamic Sandbox

Initializing Workspace

A true-or-false problem is often won by identifying the word that is too strong. Sylow theory guarantees existence, but normality and uniqueness need additional counting information.

n_p

Dynamic Sandbox

Initializing Workspace

The calculator gives data, not a full proof. A valid true-or-false answer still needs a sentence explaining which theorem applies or which counterexample satisfies the hypothesis and fails the conclusion.

EXAMPLE

p

SOLUTION

\begin{equation*} G=S_3. \end{equation*}

EXAMPLE

39

SOLUTION

|G|=39=3\cdot 13

EXAMPLE

65

SOLUTION

\begin{equation*} 65=5\cdot 13 \end{equation*}

EXAMPLE

76

SOLUTION

|G|=76=4\cdot 19

SOLVED PROBLEM

33

SOLUTION

|G|=33=3\cdot 11

SOLVED PROBLEM

G

SOLUTION

p

EXERCISE

77

ANSWER

77=7\cdot 11

3 Questions

Students often confuse existence of a Sylow subgroup with normality. Sylow guarantees existence, not uniqueness.

19

It must satisfy the hypothesis of the statement and fail the conclusion.