Let
N
∗
\mathcal N_\ast
denote the unoriented cobordism ring. Let
G
=
(
Z
/
2
)
n
G=(\mathbb Z/2)^n
and let
Z
∗
(
G
)
Z_\ast (G)
denote the equivariant cobordism ring of smooth manifolds with smooth
G
G
-actions having finite stationary points. In this paper we show that the unoriented cobordism class of the (real) flag manifold
M
=
O
(
m
)
/
(
O
(
m
1
)
×
⋯
×
O
(
m
s
)
)
M=O(m)/(O(m_1)\times \dots \times O(m_s))
is in the subalgebra generated by
⨁
i
>
2
n
N
i
\bigoplus _{i>2^n}\mathcal N_i
, where
m
=
∑
m
j
m= \sum m_j
, and
2
n
−
1
>
m
≤
2
n
2^{n-1}>m\le 2^n
. We obtain sufficient conditions for indecomposability of an element in
Z
∗
(
G
)
Z_\ast (G)
. We also obtain a sufficient condition for algebraic independence of any set of elements in
Z
∗
(
G
)
Z_\ast (G)
. Using our criteria, we construct many indecomposable elements in the kernel of the forgetful map
Z
d
(
G
)
→
N
d
Z_d(G)\to \mathcal N_d
in dimensions
2
≤
d
>
n
2\le d>n
, for
n
>
2
n>2
, and show that they generate a polynomial subalgebra of
Z
∗
(
G
)
Z_\ast (G)
.