At one time(,) I suspected that my girlfriend Hitomi was in love with my elder brother.
Do I need that comma? I ask this because I see sentences with and without on Google.
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The comma is not required. Many writers do place a comma after adjuncts which which are ‘fronted’ like the one in your sentence, but others hold that a comma here would mark the preceding phrase as ‘supplemental’ and should not be employed when the phrase acts as an adjunct modifying the clause.
If you’re under authority—in school, or taking an exam—follow the prejudices of your teacher or examiner. When you’re out from under, follow your own inclination.