Vtyjkyuk

Let’s define $sigma(G)$ as the sum of orders of all normal subgroups of a finite group $G$. Do there exist two finite groups $H$ and $K$ such, that $sigma(H) = |H| + |K| = sigma(K)$ and $H$ is non-cyclic?

Why $H$ is required to be non-cyclic? A pair of cyclic groups $H$ and $K$ satisfies that condition iff $|H|$ and $|K|$ form an amicable pair. And it would be interesting to know, what happens if at least one of those groups is non-cyclic.

A quick MAGMA code I wrote finds the following example with $$H=D_10 quad , K=D_19$$ Indeed, in both cases
$sigma(K)=1+19+38=58$
$sigma(H)=1+2+5+10+10+10+20= 58$

and $|H|+|K|=20+38 =58$.

Here is the very naive and probably bugged (the $i^j$ indexing definitely doesn't work, $2^4 = 4^2$ so I am missing some examples, and it checks everything twice which is not efficient) code if you are interested. I checked for examples up to $60$.

N:=60;
A:=AssociativeArray();
A[1]:=1;
for i in [2..N] do
w:=NumberOfSmallGroups(i);
for j in [1..w] do
G:=SmallGroup(i,j);
S:=NormalSubgroups(G);
h:=0;
for k in [1..#S] do
h:=h+Order(S[k]`subgroup);
end for;
A[i^j]:=h;
end for;
end for;
for i in [2..N] do
w:=NumberOfSmallGroups(i);
for j in [1..w] do
for a in [2..N] do
y:=NumberOfSmallGroups(a);
for b in [1..y] do
if A[i^j] eq i+a then
if A[a^b] eq i+a then print "(",i,j,")","(",a,b,")", A[i^j], A[a^b];
end if; end if; end for; end for; end for; end for;

And the output

( 12 1 ) ( 12 1 ) 24 24
( 20 4 ) ( 38 1 ) 58 58
( 24 1 ) ( 28 1 ) 52 52
( 28 1 ) ( 24 1 ) 52 52
( 28 2 ) ( 28 2 ) 56 56
( 30 1 ) ( 30 1 ) 60 60
( 38 1 ) ( 20 4 ) 58 58
( 56 1 ) ( 56 1 ) 112 112

Note that it also contains other examples that might be interesting of "self-amicable groups", such as SmallGroup(30,1) $= C_5 times S_3$.

By self-amicable I mean that they mimic the behaviour of $28$ which forms an amicable pair with itself (realised in group theory by $C_28$).

Edit: found this paper https://arxiv.org/pdf/math/0104012.pdf

Edit 2: found this paper as well http://cage.ugent.be/~tdemedts/preprints/leinster.pdf