
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In the questions below:  $G$  is a ﬁnite group;  $H \leq G$  a subgroup of  $G; |G : H |$  the index of  $H$  in  $G; |X |$  the number of elements of  $X \subseteq G; Z (G)$  the center of  $G; G'$  the commutator subgroup of  $G; N_{G}(H )$  the normalizer of  $H$  in  $G; C_{G}(H )$  the centralizer of  $H$  in  $G$ ; and  $S_{n}$  the  $n$ -th symmetric group.

Let  $H \leq G, |H | = 3.$  What can be said about  $|N_{G}(H ) : C_{G}(H )|$ ?

Present the answer in LaTex format: \boxed{Your answer}
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