Suppose if a user is a member of the Domain Admins group, Did he able to log on to the network even when a global catalog is not available?

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The global catalog is designed to respond to queries about objects anywhere in the forest with maximum speed and minimum network traffic. Because a single global catalog contains information about objects in all domains in the forest, a query about an object can be resolved by a global catalog in the domain in which the query is initiated. Thus, finding information in the directory does not produce unnecessary query traffic across domain
boundaries.
You can optionally configure any domain controller to host a global catalog, based on your organization's requirements for servicing logon requests and search queries. After additional domain controllers are installed in the domain, you can change the default location of the global catalog to another domain controller using Active Directory Sites and Services.
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