In the IoT Greengrass 1.x version, a Greengrass group is an entity which includes a Greengrass core, devices connecting to the core, connectors, lambda functions, subscriptions, and related resources that you define. You apply settings such as logging, monitoring, IAM role etc., to the Greengrass group. The Greengrass group is then deployed to the Greengrass core device.
Think of Greengrass core as the physical device that runs the Greengrass software at the edge. Thus, for each Greengrass group, you need a GreenGrass core device that is registered in AWS IoT core as a thing. Greengrass core authenticates to AWS IoT core using a X.509 certificate and has a shadow like other devices that connect directly to AWS IoT Core. But additionally, it can manage the processes needed to run locally as part of the Greengrass group.
For differences in the IoT Greengrass 2.x version, see https://docs.aws.amazon.com/greengrass/v2/developerguide/move-from-v1.html to learn how these concepts have evolved.
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