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Simple groups and irreducible characters of degree 3
See \S 8.5 in W. Feit, The current situation in the theory of finite simple groups, Actes. Cobgr. Internat. Me\ath. Nice 1970, vol.1 Gauthhier-Villars, Paris, 1971, 55-93.
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p-group with large center
In the my post above such $p$-groups are characterized so other atgument are superfluous. Indeed, let $|G:{\rm Z}(G)|\p^2$ and $S\le G$ minimal nonabelian. Set $H=S\text{Z}(G)$. As $|H:\text{Z}(G)|\ge p^2=|G:\text{Z}(G)|$, we get $H=G$. Conversely, if $G=S\text{Z}(G)$, where $S$ is minimal nonabelian, then $\text{Z}(S)\le\text{Z}(G)$ and we conclude that $|G:\text{Z}(G)|=|S:\text{Z}(S)|=p^2$.
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p-group with large center
In my post above such groups are characterized. So all other argumant are auperfluous. Indeed, let $|G:\text{Z}(G)|=p^2$ and let $S\le G$ be minimal nonabelian. Set $H=S\text{Z}(G)$. As $H$ is nonabelian, $|H:\text{Z}(G)|\ge
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Number of Normal subgroups In a p-Group
What you mean by
counting of normal subgroups' and where you offer to use this? Next, the number of maximal subgroups of a $p$-group depends not of its order but of minimal number of its generators.v All this is presented in any exposition of $p$-groups. See, for example, the first pages in Kap III of Huppert's
Endliche Gruppen'.
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Groups with all subgroups normal
In fact, Otto Schmidt (1926) has classified the finite groups all of whose nonnormal subgroups are conjugate. Later, in 1938, he also has classified the finite groups with exactly two classes of nonnormal subgroups.
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Groups with all normal subgroups characteristic
(1) I'd like to recall the old problem of study of the $p$-groups all of whose maximal subgroups are characteristic. (2) Does there exist a noncyclic group of exponent $p$, all of whose maximal subgroups are characteristic? (3) Is it true that if all subgroups of order $p$ from $\text{Z}(G)$ are characteristic in the $p$-group $G$, then $|\Omega_1(\text{Z}(G))|$ is bounded?
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5/8 bound in group theory
See also Chapter 11 in Berkovich-Zhmud', Characters of Finite Groups, Part 1.
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Finite solvable groups with abelian Fitting subgroup
Obviously, $G$ is supersolvable iff all indices of a chief series of $G$ containing $F(G)$ and lying below $F(G)$ are primes (i.e., $F(G)$ is supersolvable immersed in $G$, in terminology of R. Baer),
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Classification of automorphism groups of groups of order $p^4$
For $p=2$, si Hall-Senior, Atlas of groups of order $\le2^6$.
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classification of $p$-groups
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Approve
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On a problem of Berkovich
W. Gasch\"utz has proved that if a $p$-group $G$ is of order $>p$, then $p$ divides $|\text{Aut}(G):\text{Inn}(G)|$. Therefore, the above question is natural.