Dear Experts, 


I'm a graduate student, dealing with group-theory. 


In my current research, I used the bound "Alexander Gruber" wrote about in this post:
[See Here](https://mathoverflow.net/questions/108581/number-of-normal-subgroups-in-a-p-group)
(Actually, I have just found out that the bound is known and already mentioned online). 


I was able to give a detailed proof for this claim myself of course , but since it seems like this bound is well known, I want to add a reference . 


So... does anyone know about a reference for the bound Alexander Gruber gives in the post above? I want to add it to the Bibliography section. 


Thanks for your help! 



Normal subgroups In a p-group [Reference?]
For general Hall Algebra,


Ian MacDonald's *Symmetric Functions and Hall Polynomials*
<http://books.google.com.hk/books/about/Symmetric_Functions_and_Hall_Polynomials.html?id=srv90XiUbZoC>


For general reference about the number of vector subspaces over a finite p-field, 


I suggest that you can google the keyword "*subspaces over a finite p-field*", here're some papers and some related problems arise in StackExchange:


<https://math.stackexchange.com/questions/70801/how-many-k-dimensional-subspaces-there-are-in-n-dimentional-vector-space-over>


And a popular ref. is the Goldman's paper which may contain the bounds:


J. Goldman and G.-C. Rota, The number of subspaces of a vector space,
in \Recent progress in combinatorics" (W. Tutte, Ed.), pp. 75{83. Aca-
demic Press, San Diego, CA, 1969.


