First of all the white arrow is the Direct Selection tool which does not select a whole object just individual points. Use the regular selection tool (black arrow) to select and move a grouped object. Secondly why are you working on the outside pasteboard rather than on the artboard? However, even if you are working on the pasteboard objects should not disappear like that. Is this happening on just this file or many or all files? What version of Illustrator and what operating system are you using? If the problem is rampant then I would try resetting the program preferences just to make sure that it is not a result of program corruption. This will reset the program to its defaults which usually can fix a multitude of issues.
To reset preferences:
For Macintosh Users: The User Library folder in which Illustrator’s preferences are stored is hidden by default on most Macintoshes. To access it make sure that Illustrator is closed and click on the desktop to launch a Finder Window (Command-N).With this window in column view follow the path User>Home folder (it’s the folder with an icon that looks like a house—it may have the user’s name rather than “Home”) and click on the Home folder. With the Option Key pressed choose Library from the Finder Go Menu. “Library” will now appear within the Home folder. Within the Library folder find the folder called Preferences and within it find the folder called “Adobe Illustrator <Version #> Settings” (earlier versions of Illustrator might just say “Adobe Illustrator”) and the file called “com.adobe.Illustrator.plist” and delete both that folder and that file. When Illustrator is next launched it will create new preference files and the program will be restored to its defaults.
For Windows Users: You can try the quick way of resetting on a PC which is to hold down Ctrl + Alt + Shift when launching Illustrator and respond affirmatively when asked if you want to reset. There have been some recent reports that the window asking if you want to reset is not popping up but that the prefs are being reset anyway. If this works great but if it doesn’t you may have to manually delete them.
To do so: On Windows 7 and above the preference files are hidden. To find them go to the Control Panel and open Folder Options and then click the View tab. Then select “Show hidden files and folders” or “Show hidden files, folders or drive options” in Advanced Settings. Then delete (or rename) the folder at the end of this path: C:\Users\[User Name]\AppData\Roaming\Adobe\Adobe Illustrator [version number]\ Settings\<Language>. Make sure that Illustrator is closed when you do this. When you relaunch the program it will create new preference files and the program will be at its default settings.
The advantage of manually deleting preference files is that after you’ve reset up the program (make sure that no document window is open) to your liking, you can create copies of your personalized “mint” preference files (make sure that you quit the program before copying them—that finalizes your customization) and use them in the future to replace any corrupt versions you may need to delete.
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It looks like there is a clipping mask on that graphic. Switch to outline view and look for a regrangle surrounding all four figures. Select it and delete using object select (white arrow with +).