The state, relationship, or bond of being brothers; a community or group bound together by brotherly affection.
From 'brother' plus the suffix '-ship', which creates nouns referring to states, relationships, or groups (friendship, membership, leadership). The '-ship' suffix comes from Old English and is highly productive.
The '-ship' suffix is ancient and still thriving—we add it to new words constantly (influencer → influencership), and older formations like 'brotherhood' and 'brothership' show how different generations of English speakers made the same suffix do overlapping work.
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