Apple Inc. executive Scott Forstall was asked to leave the company after he refused to sign his name to a letter apologizing for shortcomings in Apple's new mapping service, according to people familiar with the matter.

The incident was the latest clash between Mr. Forstall, who oversaw Apple's mobile software unit, and other executives at the company. It led to one of the most significant management shake-ups in Apple's recent history and its most sweeping changes under Chief Executive Tim Cook.

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