I've heard a non-native English speaker use the expression "in this place, having a long arm to scratch someone else's back is more important than having a long résumé", conveying the idea of favoritism in a place influencing the promotion of people within. I would like to know if there is an equivalent in English to such expression.
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seems to have similar connotations, although is more about knowing the right people, than currying favour with them.
There is the more salacious
which takes the back scratching a little more literally.
I understood your question as asking for an idiom for nepotism or sycophantic promotion.
One could say, "When it comes to promotions in this company, flattery will get you everywhere."
There's a theory on the origin of "Bob's your uncle" which is based on nepotism (although the saying doesn't mean that anymore).
At first I misinterpreted the quotation, thinking it to mean, "it's more important to help other people than to look after yourself;" that is, employees should be looking out for others – particularly others within the company. In the U.S., many quotes promoting such an unselfish attitude are borrowed from the world of sports, such as:
but the most famous I thought of was:
Getting back to your question, some have parodied that expression, to put it more in line with the underhanded dealings of a workplace such as you describe:
implying that, at the end of the day, you've got to "look out for #1."
It's not a full maxim, and I think it may be an Americanism, but the verb phrase buck for can be used:
All that said, I don't think any of these are as apt as Matt's "it's not what you know, but who you know" suggestion.