Notation for formal language

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Let $\Sigma = \{\alpha, \beta \}$ and $L = \{\alpha (\beta \alpha)^*\}$

I don't understand the notation for the language $L$. Does it mean that every string must start with $\alpha$, and then have an arbitrary number of ($\beta\alpha$) after it?

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That's exactly right. And the 'arbitrary number' is 0 or more. So, just having $\alpha$ would be a word in this language.