Based on the principle of LMX theory, what observation would you make about Carly’s leadership at Mills , Smith and Peter.
The LMX theory is a theory that focuses on the interaction between the leader and the follower. Based on the case study, I observed that Carly used the in-group leadership style at Mills, Smith and Peter. She gave more attention to her preferred team and cause the other team members to be unhappy. She may be considered an ineffective leader because for better result on task to be achieved she needs to all the teams equal attention.

Is there an in-group and out-group, and, if so, which are they?
In the case study, there are both in-groups and out-groups. Jack and his team fall in to the in-group category while Terri, Julie and Sarah fall in the category of the out group. They fall under this category because they have observed that Carly has picked Jack as her favorite and they are unhappy about it.

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In what way is Carly’s relationship with the four groups productive or counterproductive to the overall goals of the agency?
Carly’s relationship with four teams is both productive and counterproductive. Jack’s team is productive because they are willing to go the extra mile to get the work done for Carly . However the other three teams Terri, Julie and Sarah are counterproductive they are unhappy and are likely not going to go the extra mile for Carly. But, I also observed that Sarah may fall under the category of productive group because she may have observed some of these favoritism by Carly but still gets the work done regardless.
Do you think Carly should change her approach toward the associate directors? If so, what should she do differently?
From the case study , the company is considered as one company. Therefore, Carly should change her approach towards the associate directors to enable them achieve more for the organization.