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My question is motivated by a recent discussion of the Frattini subgroup with my professor, and in particular, the special structure of $G/\Phi(G)$.

Given a group $G$ with normal subgroups $H_1, H_2, \ldots H_n$ there exists a natural homomorphism

$$\phi: G \to \prod_{i=1}^n \frac{G}{H_i}$$

given by $\phi(g) = (gH_1, \ldots, gH_n)$. The kernel of this map is the set of all $g$ such that $g \in H_1, \ldots g\in H_n$, i.e. $\cap_{i=1}^n H_i$. By the first isomorphism theorem we thus have that

$$ \frac{G}{\bigcap_{i=1}^n H_i} \cong \phi[G] = \{(gH_1, \ldots gH_n) \mid g \in G\}$$

Is there anything more that we can say about the structure of this group? In the context of the Frattini subgroup, the intersection of all maximal subgroups, since each $G/H_i$ is congruent to $\mathbb{Z}_p$ we have that $G/\Phi(G)$ is a product of prime-order cyclic groups. Are there other nice results about the structure of this group?

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  • $\begingroup$ When you say $G/H_i\cong C_p$ are you assuming $G$ is a $p$-group? $\endgroup$
    – anon
    Nov 1, 2014 at 18:24
  • $\begingroup$ Yes, sorry, for that to hold $G$ does have to be a $p$-group. $\endgroup$ Nov 2, 2014 at 13:16

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