Hello,
If you're just closing the browser window, you're not actually quitting the application. You need to quit the application for Flash Player to install. There are a few different ways to Quit Safari:
Quit Safari app from the Dock icon:
Mouse over the Safari icon in the Dock
Press CTRL and left-mouse click at the same time
A menu will display. Click Quit to terminate the application.
Use Activity Monitor to terminate Safari:
Launch Activity Monitor from Applications/Utilities/Activity Monitor
Ensure 'All Processes' are being listed (menu in the upper right)
Find and select Safari
Click the Quit Process button (upper left)
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Maria