As Danman wrote above, Saundra says El-jins in her Christmas message [on Ultimate Motown Christmas] and she would know!
I always thought it was pronounced El-jins though I can't even say why. For a long time I think I only saw the group's name written but never heard anyone say it.
I can understand some uncertainty, especially for people outside the US. Elgin is pronounced El-gin [with a hard g] with regards to Lord Elgin and the Elgin Marbles, as in this BBC documentary. [I have also heard them called the El-jin Marbles but maybe that was by an American anchor reading a teleprompter on CNN?]
There's a town in Scotland called Elgin, pronounced El-gin with a hard g. A Chicago Tribune article cites a historian who says that the city of Elgin, Illinois [mentioned in posts above] was named after [a song about] this town in Scotland.
https://www.chicagotribune.com/news/...121-story.html
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