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Here is a group theoretic phrasing of a special case of the union closed conjecture:

Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half the subgroups of $G$?

Motivation: Frankl's union closed sets conjecture has an equivalent phrasing in terms of lattices. It says that in every finite lattice there is a join irreducible element which is less than or equal to at most half the elements in the lattice.

Finite lattices are always isomorphic to intervals of subgroups $[H,G]$ for groups $H,G$ (i.e. the lattice of subgroups $H\subseteq K \subseteq G$, with the subgroup relation). It is not known whether it suffices to take $H$ and $G$ to be finite. I wonder if anything about Frankl's conjecture is known for the case when $H$ is the trivial group. And that is precisely what is asked above. Notice that the elements of prime power order are in correspondence with the join irreducibles of the lattice of subgroups of $G$.

Is the answer to the question above known? Is this known for special classes of groups?

Here is a group theoretic phrasing of a special case of the union closed conjecture:

Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half the subgroups of $G$?

Motivation: Frankl's union closed sets conjecture has an equivalent phrasing in terms of lattices. It says that in every finite lattice there is a join irreducible element which is less than or equal to at most half the elements in the lattice.

Finite lattices are always isomorphic to intervals of subgroups $[H,G]$ for groups $H,G$ (i.e. the lattice of subgroups $H\subseteq K \subseteq G$, with the subgroup relation). It is not known whether it suffices to take $H$ and $G$ to be finite. I wonder if anything about Frankl's conjecture is known for the case when $H$ is the trivial group. And that is precisely what is asked above. Notice that the elements of prime power order are in correspondence with the join irreducibles of the lattice of subgroups of $G$.

Is the answer to the question above known? Is this known for special classes of groups?

Here is a group theoretic phrasing of a special case of the union closed conjecture:

Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half the subgroups of $G$?

Motivation: Frankl's union closed sets conjecture has an equivalent phrasing in terms of lattices. It says that in every finite lattice there is a join irreducible element which is less than or equal to at most half the elements in the lattice.

Finite lattices are always isomorphic to intervals of subgroups $[H,G]$ for groups $H,G$ (i.e. the lattice of subgroups $H\subseteq K \subseteq G$, with the subgroup relation). It is not known whether it suffices to take $H$ and $G$ to be finite. I wonder if anything is known for the case when $H$ is the trivial group. And that is precisely what is asked above. Notice that the elements of prime power order are in correspondence with the join irreducibles of the lattice of subgroups of $G$.

Is this known? Is this known for special classes of groups?

Here is a group theoretic phrasing of a special case of the union closed conjecture:

Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half the subgroups of $G$?

Motivation: Frankl's union closed sets conjecture has an equivalent phrasing in terms of lattices. It says that in every finite lattice there is a join irreducible element which is less than or equal to at most half the elements in the lattice.

Finite lattices are always isomorphic to intervals of subgroups $[H,G]$ for groups $H,G$ (i.e. the lattice of subgroups $H\subseteq K \subseteq G$, with the subgroup relation). It is not known whether it suffices to take $H$ and $G$ to be finite. I wonder if anything about Frankl's conjecture is known for the case when $H$ is the trivial group. And that is precisely what is asked above. Notice that the elements of prime power order are in correspondence with the join irreducibles of the lattice of subgroups of $G$.

Is the answer to the question above known? Is this known for special classes of groups?